eMath:a - Κατεύθυνση
ematha, ιστότοπος μαθηματικών ασκήσεων
https://ematha.vassiliadis.edu.gr/2012-11-27-12-21-20/2012-11-27-19-25-06
2024-03-29T08:02:59+02:00
eMath:a
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21. Β΄Λυκείου/ Κατεύθυνση/ Συντεταγμένες διανύσματος
2018-01-20T15:01:43+02:00
2018-01-20T15:01:43+02:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/211-21-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CE%B5%CF%82-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%BF%CF%82
Παρίσης Γιώργος
parisis@vas.edu.gr
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to \,\,\,\kappa \alpha \iota \,\,\, \beta \limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaiiaacqWFXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugObiaa+P7acaGFXoGaa4xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqWFYoGyaSqabeaacqGHsgIRaaaaaa!4BBD!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span> με γωνία <img class="ee_img tr_noresize" alt="\left( { \alpha \limits^ \to \, ,\limits^{\widehat {}} \, \beta \limits^ \to } \right) = \phi " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20%2C%5Climits%5E%7B%5Cwidehat%20%7B%7D%7D%20%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%20%3D%20%5Cphi%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWfGaqaaGGaaiab-f7aHbWcbeqaaiabgkziUcaakiaaykW7daWfGaqaaiaacYcaaSqabKazbaiabaWcdaqiaaqcKfaGaeaaaiaawkWaaaaakiaaykW7daWfGaqaaiab-j7aIbWcbeqaaiabgkziUcaaaOGaayjkaiaawMcaaiabg2da9iab-v9aMbaa!48C9!" style="vertical-align: middle;" /></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;">καθώς και τα διαφορετικά ανά δύο σημεία <img class="ee_img tr_noresize" alt="{\rm A}\left( {2\eta \mu \varphi \,,\,\sigma \upsilon \nu \varphi } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Crm%20A%7D%5Cleft%28%20%7B2%5Ceta%20%5Cmu%20%5Cvarphi%20%5C%2C%2C%5C%2C%5Csigma%20%5Cupsilon%20%5Cnu%20%5Cvarphi%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKde0aaeWaaeaaiiaacqWFYaGmcqWF3oaAcqWF8oqBcqWFgpGAcaaMc8UaaiilaiaaykW7ieaacaGFdpGaa4xXdiaa+17acqWFgpGAaiaawIcacaGLPaaaaaa!4837!" style="vertical-align: middle;" />, <img class="ee_img tr_noresize" alt="{\rm B}\left( { - \sigma \upsilon \nu \varphi \,,\,\eta \mu \varphi } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Crm%20B%7D%5Cleft%28%20%7B%20-%20%5Csigma%20%5Cupsilon%20%5Cnu%20%5Cvarphi%20%5C%2C%2C%5C%2C%5Ceta%20%5Cmu%20%5Cvarphi%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaeWaaeaacqGHsislieaacaWFdpGaa8xXdiaa-17aiiaacqGFgpGAcaaMc8UaaiilaiaaykW7cqGF3oaAcqGF8oqBcqGFgpGAaiaawIcacaGLPaaaaaa!483C!" style="vertical-align: middle;" /> και <img class="ee_img tr_noresize" alt="\Gamma \left( {\sigma \upsilon \nu \varphi \,,\,\eta \mu \varphi } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5CGamma%20%5Cleft%28%20%7B%5Csigma%20%5Cupsilon%20%5Cnu%20%5Cvarphi%20%5C%2C%2C%5C%2C%5Ceta%20%5Cmu%20%5Cvarphi%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaeWaaeaaieaacaWFdpGaa8xXdiaa-17aiiaacqGFgpGAcaaMc8UaaiilaiaaykW7cqGF3oaAcqGF8oqBcqGFgpGAaiaawIcacaGLPaaaaaa!4755!" style="vertical-align: middle;" />.</span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\phi \in \left[ {0,\frac{\pi }{2}} \right) \cup \left( {\frac{\pi }{2},\pi } \right]" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cphi%20%20%5Cin%20%5Cleft%5B%20%7B0%2C%5Cfrac%7B%5Cpi%20%7D%7B2%7D%7D%20%5Cright%29%20%5Ccup%20%5Cleft%28%20%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D%2C%5Cpi%20%7D%20%5Cright%5D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8x1dyMaeyicI48aaKGeaeaacaaIWaGaaiilamaalaaabaGae8hWdahabaGaaGOmaaaaaiaawUfacaGLPaaacqGHQicYdaqcWaqaamaalaaabaGae8hWdahabaGaaGOmaaaacaGGSaGae8hWdahacaGLOaGaayzxaaaaaa!4786!" style="vertical-align: middle;" /><span style="position: relative; top: 14pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> Β. Αν τα σημεία αυτά είναι συνευθειακά να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\varepsilon \phi \phi = 1" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvarepsilon%20%5Cphi%20%5Cphi%20%20%3D%201" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8xTduMae8x1dyMae8x1dyMaeyypa0JaaGymaaaa!3CE9!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> Γ. Αν <img class="ee_img tr_noresize" alt="\left| { {{\rm A}{\rm B}}\limits^ \to } \right| = \left| { {{\rm A}\Gamma }\limits^ \to } \right|\," src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%7B%7B%5Crm%20A%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%5C%2C" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabfg5abjabfk5acbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqqHroqqcqqHtoWraSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacaaMc8oaaa!48D4!" style="vertical-align: middle;" /><span style="position: relative; top: 14pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> 1. Να αποδειχτεί ότι: <img class="ee_img tr_noresize" alt="\, \alpha \limits^ \to \,|\,\,| \beta \limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%7C%5C%2C%5C%2C%7C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVpaaxacabaaccaGae8xSdegaleqabaGaeyOKH4kaaOGaaGPaVlaacYhacaaMc8UaaGPaVlaacYhadaWfGaqaaiab-j7aIbWcbeqaaiabgkziUcaaaaa!45D7!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> 2. Αν επιπλέον<span style="position: relative; top: 5pt;"> </span><img class="ee_img tr_noresize" alt="{ {\, \alpha \limits^ \to \, \uparrow \downarrow \, \beta \limits^ \to }\limits_{} }" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%20%7B%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20%5Cuparrow%20%20%5Cdownarrow%20%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%5Climits_%7B%7D%20%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcbaEaamaaxababaGaaGPaVpaaxacabaaccaGae8xSdegajia4beqaaiabgkziUcaajea4caaMc8UaeyyKH0Qaey4KH8QaaGPaVpaaxacabaGae8NSdigajia4beqaaiabgkziUcaaaeaaaeqaaaaa!49D4!" style="vertical-align: middle;" /> να υπολογιστεί η γωνία θ του <img class="ee_img tr_noresize" alt=" {\Gamma {\rm A}}\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%5CGamma%20%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHtoWrcqqHroqqaSqabeaacqGHsgIRaaaaaa!3AF3!" style="vertical-align: middle;" /><span style="position: relative; top: 2pt;"> </span> με τον ημιάξονα Οx.</span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to \,\,\,\kappa \alpha \iota \,\,\, \beta \limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaiiaacqWFXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugObiaa+P7acaGFXoGaa4xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqWFYoGyaSqabeaacqGHsgIRaaaaaa!4BBD!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span> με γωνία <img class="ee_img tr_noresize" alt="\left( { \alpha \limits^ \to \, ,\limits^{\widehat {}} \, \beta \limits^ \to } \right) = \phi " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20%2C%5Climits%5E%7B%5Cwidehat%20%7B%7D%7D%20%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%20%3D%20%5Cphi%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWfGaqaaGGaaiab-f7aHbWcbeqaaiabgkziUcaakiaaykW7daWfGaqaaiaacYcaaSqabKazbaiabaWcdaqiaaqcKfaGaeaaaiaawkWaaaaakiaaykW7daWfGaqaaiab-j7aIbWcbeqaaiabgkziUcaaaOGaayjkaiaawMcaaiabg2da9iab-v9aMbaa!48C9!" style="vertical-align: middle;" /></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;">καθώς και τα διαφορετικά ανά δύο σημεία <img class="ee_img tr_noresize" alt="{\rm A}\left( {2\eta \mu \varphi \,,\,\sigma \upsilon \nu \varphi } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Crm%20A%7D%5Cleft%28%20%7B2%5Ceta%20%5Cmu%20%5Cvarphi%20%5C%2C%2C%5C%2C%5Csigma%20%5Cupsilon%20%5Cnu%20%5Cvarphi%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKde0aaeWaaeaaiiaacqWFYaGmcqWF3oaAcqWF8oqBcqWFgpGAcaaMc8UaaiilaiaaykW7ieaacaGFdpGaa4xXdiaa+17acqWFgpGAaiaawIcacaGLPaaaaaa!4837!" style="vertical-align: middle;" />, <img class="ee_img tr_noresize" alt="{\rm B}\left( { - \sigma \upsilon \nu \varphi \,,\,\eta \mu \varphi } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Crm%20B%7D%5Cleft%28%20%7B%20-%20%5Csigma%20%5Cupsilon%20%5Cnu%20%5Cvarphi%20%5C%2C%2C%5C%2C%5Ceta%20%5Cmu%20%5Cvarphi%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaeWaaeaacqGHsislieaacaWFdpGaa8xXdiaa-17aiiaacqGFgpGAcaaMc8UaaiilaiaaykW7cqGF3oaAcqGF8oqBcqGFgpGAaiaawIcacaGLPaaaaaa!483C!" style="vertical-align: middle;" /> και <img class="ee_img tr_noresize" alt="\Gamma \left( {\sigma \upsilon \nu \varphi \,,\,\eta \mu \varphi } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5CGamma%20%5Cleft%28%20%7B%5Csigma%20%5Cupsilon%20%5Cnu%20%5Cvarphi%20%5C%2C%2C%5C%2C%5Ceta%20%5Cmu%20%5Cvarphi%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaeWaaeaaieaacaWFdpGaa8xXdiaa-17aiiaacqGFgpGAcaaMc8UaaiilaiaaykW7cqGF3oaAcqGF8oqBcqGFgpGAaiaawIcacaGLPaaaaaa!4755!" style="vertical-align: middle;" />.</span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\phi \in \left[ {0,\frac{\pi }{2}} \right) \cup \left( {\frac{\pi }{2},\pi } \right]" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cphi%20%20%5Cin%20%5Cleft%5B%20%7B0%2C%5Cfrac%7B%5Cpi%20%7D%7B2%7D%7D%20%5Cright%29%20%5Ccup%20%5Cleft%28%20%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D%2C%5Cpi%20%7D%20%5Cright%5D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8x1dyMaeyicI48aaKGeaeaacaaIWaGaaiilamaalaaabaGae8hWdahabaGaaGOmaaaaaiaawUfacaGLPaaacqGHQicYdaqcWaqaamaalaaabaGae8hWdahabaGaaGOmaaaacaGGSaGae8hWdahacaGLOaGaayzxaaaaaa!4786!" style="vertical-align: middle;" /><span style="position: relative; top: 14pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> Β. Αν τα σημεία αυτά είναι συνευθειακά να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\varepsilon \phi \phi = 1" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvarepsilon%20%5Cphi%20%5Cphi%20%20%3D%201" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8xTduMae8x1dyMae8x1dyMaeyypa0JaaGymaaaa!3CE9!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> Γ. Αν <img class="ee_img tr_noresize" alt="\left| { {{\rm A}{\rm B}}\limits^ \to } \right| = \left| { {{\rm A}\Gamma }\limits^ \to } \right|\," src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%7B%7B%5Crm%20A%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%5C%2C" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabfg5abjabfk5acbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqqHroqqcqqHtoWraSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacaaMc8oaaa!48D4!" style="vertical-align: middle;" /><span style="position: relative; top: 14pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> 1. Να αποδειχτεί ότι: <img class="ee_img tr_noresize" alt="\, \alpha \limits^ \to \,|\,\,| \beta \limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%7C%5C%2C%5C%2C%7C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVpaaxacabaaccaGae8xSdegaleqabaGaeyOKH4kaaOGaaGPaVlaacYhacaaMc8UaaGPaVlaacYhadaWfGaqaaiab-j7aIbWcbeqaaiabgkziUcaaaaa!45D7!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"></span></span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="font-size: 12pt; color: #000000;"> 2. Αν επιπλέον<span style="position: relative; top: 5pt;"> </span><img class="ee_img tr_noresize" alt="{ {\, \alpha \limits^ \to \, \uparrow \downarrow \, \beta \limits^ \to }\limits_{} }" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%20%7B%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20%5Cuparrow%20%20%5Cdownarrow%20%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%5Climits_%7B%7D%20%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcbaEaamaaxababaGaaGPaVpaaxacabaaccaGae8xSdegajia4beqaaiabgkziUcaajea4caaMc8UaeyyKH0Qaey4KH8QaaGPaVpaaxacabaGae8NSdigajia4beqaaiabgkziUcaaaeaaaeqaaaaa!49D4!" style="vertical-align: middle;" /> να υπολογιστεί η γωνία θ του <img class="ee_img tr_noresize" alt=" {\Gamma {\rm A}}\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%5CGamma%20%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHtoWrcqqHroqqaSqabeaacqGHsgIRaaaaaa!3AF3!" style="vertical-align: middle;" /><span style="position: relative; top: 2pt;"> </span> με τον ημιάξονα Οx.</span></p>
20. Β΄Λυκείου/ Κατεύθυνση/ Συντεταγμένες διανύσματος
2017-11-23T16:58:13+02:00
2017-11-23T16:58:13+02:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/207-20-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CE%B5%CF%82-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%BF%CF%82
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt; color: #000000;">Δίνονται τα σημεία <img class="ee_img tr_noresize" alt="{\rm A}\left( {\beta + 1\,,\, - \alpha } \right),\,\,\,{\rm B}\left( {\alpha + 1,\beta } \right),\,\,\,\Gamma \left( {2\alpha + 1,\alpha + 2\beta } \right),\,\,\,\alpha ,\beta \in R" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Crm%20A%7D%5Cleft%28%20%7B%5Cbeta%20%20%2B%201%5C%2C%2C%5C%2C%20-%20%5Calpha%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%7B%5Crm%20B%7D%5Cleft%28%20%7B%5Calpha%20%20%2B%201%2C%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5CGamma%20%5Cleft%28%20%7B2%5Calpha%20%20%2B%201%2C%5Calpha%20%20%2B%202%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Calpha%20%2C%5Cbeta%20%20%5Cin%20R" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeuyKde0aaeWaaeaacqaHYoGycqGHRaWkcaaIXaGaaGPaVlaacYcacaaMc8UaeyOeI0IaeqySdegacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlabfk5acnaabmaabaGaeqySdeMaey4kaSIaaGymaiaacYcacqaHYoGyaiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8Uaeu4KdC0aaeWaaeaacaaIYaGaeqySdeMaey4kaSIaaGymaiaacYcacqaHXoqycqGHRaWkcaaIYaGaeqOSdigacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlabeg7aHjaacYcacqaHYoGycqGHiiIZcaWGsbaaaa!6D1A!" style="vertical-align: middle;" /><span style="position: relative; top: 7pt;"> </span>.</span></p>
<p><span style="font-size: 12pt; color: #000000;">Αν για τα διανύσματα <img class="ee_img tr_noresize" alt=" {{\rm A}{\rm B}}\limits^ \to \,\,\,\kappa \alpha \iota \,\,\, {{\rm B}\Gamma }\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqaSqabKqaahaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugObiaa-P7acaWFXoGaa8xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHsoGqcqqHtoWraSqabKqaahaacqGHsgIRaaaaaa!5305!" style="vertical-align: middle;" /> ισχύουν <img class="ee_img tr_noresize" alt=" {{\rm A}{\rm B}}\limits^ \to \,|\,|\, {{\rm B}\Gamma }\limits^ \to \,\,\,\kappa \alpha \iota \,\,\, {{\rm A}{\rm B}}\limits^ \to \, \ne \, {{\rm B}\Gamma }\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%7C%5C%2C%7C%5C%2C%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%20%5Cne%20%5C%2C%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqaSqabKqaahaacqGHsgIRaaGccaaMc8UaaiiFaiaaykW7caGG8bGaaGPaVpaaxacabaGaeuOKdiKaeu4KdCealeqajeaWbaGaeyOKH4kaaOGaaGPaVlaaykW7caaMc8ocbaqcLbAacaWF6oGaa8xSdiaa-L7akiaaykW7caaMc8UaaGPaVpaaxacabaGaeuyKdeKaeuOKdiealeqajeaWbaGaeyOKH4kaaOGaaGPaVlabgcMi5kaaykW7daWfGaqaaiabfk5acjabfo5ahbWcbeqcbaCaaiabgkziUcaaaaa!6A23!" style="vertical-align: middle;" /> <span style="position: relative; top: 5pt;"> </span></span></p>
<p><span style="font-size: 12pt; color: #000000;">Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\alpha + \beta = 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%20%2B%20%5Cbeta%20%20%3D%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey4kaSIaeqOSdiMaeyypa0JaaGimaaaa!3F3D!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"></span></span></p>
<p><span style="font-size: 12pt; color: #000000;">Β. Αν επιπλέον ισχύουν <img class="ee_img tr_noresize" alt="\left( {{\rm O}x\,, {{\rm A}{\rm B}}\limits^ \to } \right) = {180^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%7B%5Crm%20O%7Dx%5C%2C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%20%3D%20%7B180%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqqHFoWtcaWG4bGaaGPaVlaacYcadaWfGaqaaiabfg5abjabfk5acbWcbeqcbaCaaiabgkziUcaaaOGaayjkaiaawMcaaiabg2da9iaaigdacaaI4aGaaGimamaaCaaaleqabaaccaGae83Vd8gaaaaa!4A8E!" style="vertical-align: middle;" /> και <img class="ee_img tr_noresize" alt="\left| { {{\rm A}{\rm B}}\limits^ \to } \right| = 10" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%2010" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabfg5abjabfk5acbWcbeqcbaCaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9iaaigdacaaIWaaaaa!44C4!" style="vertical-align: middle;" /></span><br /><span style="font-size: 12pt; color: #000000;"> 1. Να βρεθούν τα <img class="ee_img tr_noresize" alt="\alpha ,\beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%2C%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIbaa!3D4B!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span></span><br /><span style="font-size: 12pt; color: #000000;"> 2. Να υπολογιστεί η απόσταση των μέσων Μ και Ν των ευθύγραμμων τμημάτων ΑΒ και ΒΓ αντίστοιχα.</span></p>
<p><span style="font-size: 12pt; color: #000000;">Δίνονται τα σημεία <img class="ee_img tr_noresize" alt="{\rm A}\left( {\beta + 1\,,\, - \alpha } \right),\,\,\,{\rm B}\left( {\alpha + 1,\beta } \right),\,\,\,\Gamma \left( {2\alpha + 1,\alpha + 2\beta } \right),\,\,\,\alpha ,\beta \in R" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Crm%20A%7D%5Cleft%28%20%7B%5Cbeta%20%20%2B%201%5C%2C%2C%5C%2C%20-%20%5Calpha%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%7B%5Crm%20B%7D%5Cleft%28%20%7B%5Calpha%20%20%2B%201%2C%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5CGamma%20%5Cleft%28%20%7B2%5Calpha%20%20%2B%201%2C%5Calpha%20%20%2B%202%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Calpha%20%2C%5Cbeta%20%20%5Cin%20R" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeuyKde0aaeWaaeaacqaHYoGycqGHRaWkcaaIXaGaaGPaVlaacYcacaaMc8UaeyOeI0IaeqySdegacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlabfk5acnaabmaabaGaeqySdeMaey4kaSIaaGymaiaacYcacqaHYoGyaiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8Uaeu4KdC0aaeWaaeaacaaIYaGaeqySdeMaey4kaSIaaGymaiaacYcacqaHXoqycqGHRaWkcaaIYaGaeqOSdigacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlabeg7aHjaacYcacqaHYoGycqGHiiIZcaWGsbaaaa!6D1A!" style="vertical-align: middle;" /><span style="position: relative; top: 7pt;"> </span>.</span></p>
<p><span style="font-size: 12pt; color: #000000;">Αν για τα διανύσματα <img class="ee_img tr_noresize" alt=" {{\rm A}{\rm B}}\limits^ \to \,\,\,\kappa \alpha \iota \,\,\, {{\rm B}\Gamma }\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqaSqabKqaahaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugObiaa-P7acaWFXoGaa8xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHsoGqcqqHtoWraSqabKqaahaacqGHsgIRaaaaaa!5305!" style="vertical-align: middle;" /> ισχύουν <img class="ee_img tr_noresize" alt=" {{\rm A}{\rm B}}\limits^ \to \,|\,|\, {{\rm B}\Gamma }\limits^ \to \,\,\,\kappa \alpha \iota \,\,\, {{\rm A}{\rm B}}\limits^ \to \, \ne \, {{\rm B}\Gamma }\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%7C%5C%2C%7C%5C%2C%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%20%5Cne%20%5C%2C%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqaSqabKqaahaacqGHsgIRaaGccaaMc8UaaiiFaiaaykW7caGG8bGaaGPaVpaaxacabaGaeuOKdiKaeu4KdCealeqajeaWbaGaeyOKH4kaaOGaaGPaVlaaykW7caaMc8ocbaqcLbAacaWF6oGaa8xSdiaa-L7akiaaykW7caaMc8UaaGPaVpaaxacabaGaeuyKdeKaeuOKdiealeqajeaWbaGaeyOKH4kaaOGaaGPaVlabgcMi5kaaykW7daWfGaqaaiabfk5acjabfo5ahbWcbeqcbaCaaiabgkziUcaaaaa!6A23!" style="vertical-align: middle;" /> <span style="position: relative; top: 5pt;"> </span></span></p>
<p><span style="font-size: 12pt; color: #000000;">Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\alpha + \beta = 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%20%2B%20%5Cbeta%20%20%3D%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey4kaSIaeqOSdiMaeyypa0JaaGimaaaa!3F3D!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"></span></span></p>
<p><span style="font-size: 12pt; color: #000000;">Β. Αν επιπλέον ισχύουν <img class="ee_img tr_noresize" alt="\left( {{\rm O}x\,, {{\rm A}{\rm B}}\limits^ \to } \right) = {180^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%7B%5Crm%20O%7Dx%5C%2C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%20%3D%20%7B180%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqqHFoWtcaWG4bGaaGPaVlaacYcadaWfGaqaaiabfg5abjabfk5acbWcbeqcbaCaaiabgkziUcaaaOGaayjkaiaawMcaaiabg2da9iaaigdacaaI4aGaaGimamaaCaaaleqabaaccaGae83Vd8gaaaaa!4A8E!" style="vertical-align: middle;" /> και <img class="ee_img tr_noresize" alt="\left| { {{\rm A}{\rm B}}\limits^ \to } \right| = 10" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%2010" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabfg5abjabfk5acbWcbeqcbaCaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9iaaigdacaaIWaaaaa!44C4!" style="vertical-align: middle;" /></span><br /><span style="font-size: 12pt; color: #000000;"> 1. Να βρεθούν τα <img class="ee_img tr_noresize" alt="\alpha ,\beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%2C%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8aq0xe9Fve9Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIbaa!3D4B!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span></span><br /><span style="font-size: 12pt; color: #000000;"> 2. Να υπολογιστεί η απόσταση των μέσων Μ και Ν των ευθύγραμμων τμημάτων ΑΒ και ΒΓ αντίστοιχα.</span></p>
19. Β΄Λυκείου/ Κατεύθυνση/ Εσωτερικό γινόμενο διανυσμάτων
2017-01-06T15:26:05+02:00
2017-01-06T15:26:05+02:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/190-19-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B5%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CF%8C-%CE%B3%CE%B9%CE%BD%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF-%CE%B4%CE%B9%CE%B1%CE%BD%CF%85%CF%83%CE%BC%CE%AC%CF%84%CF%89%CE%BD
Παρίσης Γιώργος
parisis@vas.edu.gr
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;">Α. Για τα μη συγγραμμικά διανύσματα <img class="ee_img tr_noresize" alt="\vec \alpha ,\vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Calpha%20%2C%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySdeMbaSaacaGGSaGafqOSdiMbaSaaaaa!3987!" style="vertical-align: middle;" /> ισχύει <img class="ee_img tr_noresize" alt="\kappa \vec \alpha + \lambda \vec \beta = \vec 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ckappa%20%5Cvec%20%5Calpha%20%20%2B%20%5Clambda%20%5Cvec%20%5Cbeta%20%20%3D%20%5Cvec%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOUdSMafqySdeMbaSaacqGHRaWkcqaH7oaBcuaHYoGygaWcaiabg2da9iqaicdagaWcaaaa!3EF1!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span>. </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;"> Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\kappa = \lambda = 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ckappa%20%20%3D%20%5Clambda%20%20%3D%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOUdSMaeyypa0Jaeq4UdWMaeyypa0JaaGimaaaa!3B9F!" style="vertical-align: middle;" /><span style="position: relative; top: 3pt;"> </span> </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;">Β. Δίνονται τα διανύσματα <img class="ee_img tr_noresize" alt="\vec \alpha ,\vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Calpha%20%2C%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySdeMbaSaacaGGSaGafqOSdiMbaSaaaaa!3987!" style="vertical-align: middle;" /> για τα οποία ισχύουν:</span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;"> <img class="ee_img tr_noresize" alt="\left( {\vec \beta + 2\vec \alpha } \right) \bot \left( {\vec \alpha - \vec \beta } \right),\,\,\,\vec \alpha \ne \vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cvec%20%5Cbeta%20%20%2B%202%5Cvec%20%5Calpha%20%7D%20%5Cright%29%20%5Cbot%20%5Cleft%28%20%7B%5Cvec%20%5Calpha%20%20-%20%5Cvec%20%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Cvec%20%5Calpha%20%20%5Cne%20%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacuaHYoGygaWcaiabgUcaRiaaikdacuaHXoqygaWcaaGaayjkaiaawMcaaiabgwQiEnaabmaabaGafqySdeMbaSaacqGHsislcuaHYoGygaWcaaGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7cuaHXoqygaWcaiabgcMi5kqbek7aIzaalaaaaa!4E05!" style="vertical-align: middle;" /> και <img class="ee_img tr_noresize" alt="2\vec \alpha + \vec \beta \ne \vec 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=2%5Cvec%20%5Calpha%20%20%2B%20%5Cvec%20%5Cbeta%20%20%5Cne%20%5Cvec%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGOmaiqbeg7aHzaalaGaey4kaSIafqOSdiMbaSaacqGHGjsUceaIWaGbaSaaaaa!3D08!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span></span><br /><span style="color: #000000; font-size: 12pt;"> 1. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\vec \alpha ,\vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Calpha%20%2C%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySdeMbaSaacaGGSaGafqOSdiMbaSaaaaa!3987!" style="vertical-align: middle;" /> μη συγγραμμικά</span><br /><span style="color: #000000; font-size: 12pt;"> 2. Αν για το διάνυσμα <img class="ee_img tr_noresize" alt="\vec \gamma " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Cgamma%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaSaaaaa!372C!" style="vertical-align: middle;" /> ισχύει <img class="ee_img tr_noresize" alt="\left( {\vec \gamma - \vec \beta } \right)\,|\,|\,\vec \alpha \,\,\kappa \alpha \iota \,\,\,\vec \gamma \,|\,|\,\left( {\vec \alpha + 2\vec \beta } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cvec%20%5Cgamma%20%20-%20%5Cvec%20%5Cbeta%20%7D%20%5Cright%29%5C%2C%7C%5C%2C%7C%5C%2C%5Cvec%20%5Calpha%20%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%5Cvec%20%5Cgamma%20%5C%2C%7C%5C%2C%7C%5C%2C%5Cleft%28%20%7B%5Cvec%20%5Calpha%20%20%2B%202%5Cvec%20%5Cbeta%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacuaHZoWzgaWcaiabgkHiTiqbek7aIzaalaaacaGLOaGaayzkaaGaaGPaVlaacYhacaaMc8UaaiiFaiaaykW7cuaHXoqygaWcaiaaykW7caaMc8ocbaqcLbyacaWF6oGaa8xSdiaa-L7akiaaykW7caaMc8UaaGPaVlqbeo7aNzaalaGaaGPaVlaacYhacaaMc8UaaiiFaiaaykW7daqadaqaaiqbeg7aHzaalaGaey4kaSIaaGOmaiqbek7aIzaalaaacaGLOaGaayzkaaaaaa!5EF4!" style="vertical-align: middle;" /> </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;"> να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\vec \gamma = \vec \beta + \frac{1}{2}\vec \alpha " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Cgamma%20%20%3D%20%5Cvec%20%5Cbeta%20%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Cvec%20%5Calpha%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaSaacqGH9aqpcuaHYoGygaWcaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGafqySdeMbaSaaaaa!3DFF!" style="vertical-align: middle;" /><span style="position: relative; top: 12pt;"> </span> </span><br /><span style="color: #000000; font-size: 12pt;"> 3. Δίνεται ότι <img class="ee_img tr_noresize" alt="\left| {\vec \alpha } \right| = \sqrt 2 " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%5Cvec%20%5Calpha%20%7D%20%5Cright%7C%20%3D%20%5Csqrt%202%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacuaHXoqygaWcaaGaay5bSlaawIa7aiabg2da9maakaaabaGaaGOmaaWcbeaaaaa!3C23!" style="vertical-align: middle;" />. Να υπολογιστεί το μέτρο του <img class="ee_img tr_noresize" alt="\vec \gamma = \vec \beta + \frac{1}{2}\vec \alpha " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Cgamma%20%20%3D%20%5Cvec%20%5Cbeta%20%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Cvec%20%5Calpha%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaSaacqGH9aqpcuaHYoGygaWcaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGafqySdeMbaSaaaaa!3DFF!" style="vertical-align: middle;" /> </span><span style="position: relative; top: 12pt;"> </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;">Α. Για τα μη συγγραμμικά διανύσματα <img class="ee_img tr_noresize" alt="\vec \alpha ,\vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Calpha%20%2C%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySdeMbaSaacaGGSaGafqOSdiMbaSaaaaa!3987!" style="vertical-align: middle;" /> ισχύει <img class="ee_img tr_noresize" alt="\kappa \vec \alpha + \lambda \vec \beta = \vec 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ckappa%20%5Cvec%20%5Calpha%20%20%2B%20%5Clambda%20%5Cvec%20%5Cbeta%20%20%3D%20%5Cvec%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOUdSMafqySdeMbaSaacqGHRaWkcqaH7oaBcuaHYoGygaWcaiabg2da9iqaicdagaWcaaaa!3EF1!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span>. </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;"> Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\kappa = \lambda = 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ckappa%20%20%3D%20%5Clambda%20%20%3D%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqOUdSMaeyypa0Jaeq4UdWMaeyypa0JaaGimaaaa!3B9F!" style="vertical-align: middle;" /><span style="position: relative; top: 3pt;"> </span> </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;">Β. Δίνονται τα διανύσματα <img class="ee_img tr_noresize" alt="\vec \alpha ,\vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Calpha%20%2C%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySdeMbaSaacaGGSaGafqOSdiMbaSaaaaa!3987!" style="vertical-align: middle;" /> για τα οποία ισχύουν:</span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;"> <img class="ee_img tr_noresize" alt="\left( {\vec \beta + 2\vec \alpha } \right) \bot \left( {\vec \alpha - \vec \beta } \right),\,\,\,\vec \alpha \ne \vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cvec%20%5Cbeta%20%20%2B%202%5Cvec%20%5Calpha%20%7D%20%5Cright%29%20%5Cbot%20%5Cleft%28%20%7B%5Cvec%20%5Calpha%20%20-%20%5Cvec%20%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Cvec%20%5Calpha%20%20%5Cne%20%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacuaHYoGygaWcaiabgUcaRiaaikdacuaHXoqygaWcaaGaayjkaiaawMcaaiabgwQiEnaabmaabaGafqySdeMbaSaacqGHsislcuaHYoGygaWcaaGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7cuaHXoqygaWcaiabgcMi5kqbek7aIzaalaaaaa!4E05!" style="vertical-align: middle;" /> και <img class="ee_img tr_noresize" alt="2\vec \alpha + \vec \beta \ne \vec 0" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=2%5Cvec%20%5Calpha%20%20%2B%20%5Cvec%20%5Cbeta%20%20%5Cne%20%5Cvec%200" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGOmaiqbeg7aHzaalaGaey4kaSIafqOSdiMbaSaacqGHGjsUceaIWaGbaSaaaaa!3D08!" style="vertical-align: middle;" /><span style="position: relative; top: 5pt;"> </span></span><br /><span style="color: #000000; font-size: 12pt;"> 1. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\vec \alpha ,\vec \beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Calpha%20%2C%5Cvec%20%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafqySdeMbaSaacaGGSaGafqOSdiMbaSaaaaa!3987!" style="vertical-align: middle;" /> μη συγγραμμικά</span><br /><span style="color: #000000; font-size: 12pt;"> 2. Αν για το διάνυσμα <img class="ee_img tr_noresize" alt="\vec \gamma " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Cgamma%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaSaaaaa!372C!" style="vertical-align: middle;" /> ισχύει <img class="ee_img tr_noresize" alt="\left( {\vec \gamma - \vec \beta } \right)\,|\,|\,\vec \alpha \,\,\kappa \alpha \iota \,\,\,\vec \gamma \,|\,|\,\left( {\vec \alpha + 2\vec \beta } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cvec%20%5Cgamma%20%20-%20%5Cvec%20%5Cbeta%20%7D%20%5Cright%29%5C%2C%7C%5C%2C%7C%5C%2C%5Cvec%20%5Calpha%20%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%5Cvec%20%5Cgamma%20%5C%2C%7C%5C%2C%7C%5C%2C%5Cleft%28%20%7B%5Cvec%20%5Calpha%20%20%2B%202%5Cvec%20%5Cbeta%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacuaHZoWzgaWcaiabgkHiTiqbek7aIzaalaaacaGLOaGaayzkaaGaaGPaVlaacYhacaaMc8UaaiiFaiaaykW7cuaHXoqygaWcaiaaykW7caaMc8ocbaqcLbyacaWF6oGaa8xSdiaa-L7akiaaykW7caaMc8UaaGPaVlqbeo7aNzaalaGaaGPaVlaacYhacaaMc8UaaiiFaiaaykW7daqadaqaaiqbeg7aHzaalaGaey4kaSIaaGOmaiqbek7aIzaalaaacaGLOaGaayzkaaaaaa!5EF4!" style="vertical-align: middle;" /> </span></p>
<p style="margin-bottom: 8pt; line-height: 107%;"><span style="color: #000000; font-size: 12pt;"> να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\vec \gamma = \vec \beta + \frac{1}{2}\vec \alpha " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Cgamma%20%20%3D%20%5Cvec%20%5Cbeta%20%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Cvec%20%5Calpha%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaSaacqGH9aqpcuaHYoGygaWcaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGafqySdeMbaSaaaaa!3DFF!" style="vertical-align: middle;" /><span style="position: relative; top: 12pt;"> </span> </span><br /><span style="color: #000000; font-size: 12pt;"> 3. Δίνεται ότι <img class="ee_img tr_noresize" alt="\left| {\vec \alpha } \right| = \sqrt 2 " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%5Cvec%20%5Calpha%20%7D%20%5Cright%7C%20%3D%20%5Csqrt%202%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacuaHXoqygaWcaaGaay5bSlaawIa7aiabg2da9maakaaabaGaaGOmaaWcbeaaaaa!3C23!" style="vertical-align: middle;" />. Να υπολογιστεί το μέτρο του <img class="ee_img tr_noresize" alt="\vec \gamma = \vec \beta + \frac{1}{2}\vec \alpha " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cvec%20%5Cgamma%20%20%3D%20%5Cvec%20%5Cbeta%20%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Cvec%20%5Calpha%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaSaacqGH9aqpcuaHYoGygaWcaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGafqySdeMbaSaaaaa!3DFF!" style="vertical-align: middle;" /> </span><span style="position: relative; top: 12pt;"> </span></p>
18. Β΄Λυκείου/ Κατεύθυνση/ Εσωτερικό γινόμενο διανυσμάτων
2015-12-21T21:24:39+02:00
2015-12-21T21:24:39+02:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/181-18-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B5%CF%83%CF%89%CF%84%CE%B5%CF%81%CE%B9%CE%BA%CF%8C-%CE%B3%CE%B9%CE%BD%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF-%CE%B4%CE%B9%CE%B1%CE%BD%CF%85%CF%83%CE%BC%CE%AC%CF%84%CF%89%CE%BD
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt; color: #000000;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to = \left( {\alpha \,,\,2\beta + 1} \right),\,\,\, \beta \limits^ \to = \left( {4\beta - \alpha \,,\,1 - 2\beta } \right),\,\,\,\alpha ,\beta \in R" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%5C%2C%2C%5C%2C2%5Cbeta%20%20%2B%201%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B4%5Cbeta%20%20-%20%5Calpha%20%5C%2C%2C%5C%2C1%20-%202%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Calpha%20%2C%5Cbeta%20%20%5Cin%20R" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiabeg7aHjaaykW7caGGSaGaaGPaVlaaikdacqaHYoGycqGHRaWkcaaIXaaacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVpaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaaI0aGaeqOSdiMaeyOeI0IaeqySdeMaaGPaVlaacYcacaaMc8UaaGymaiabgkHiTiaaikdacqaHYoGyaiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8UaeqySdeMaaiilaiabek7aIjabgIGiolaadkfaaaa!6954!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 12pt; color: #000000;">Αν <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to \bot \beta \limits^ \to \,\,\,\kappa \alpha \iota \,\,\,\alpha > 2\beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%5Cbot%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%5Calpha%20%20%3E%202%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHLkIxdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakiaaykW7caaMc8UaaGPaVJqaaKqzagGaa8NUdiaa-f7acaWF5oGccaaMc8UaaGPaVlaaykW7cqaHXoqycqGH+aGpcaaIYaGaeqOSdigaaa!51DD!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 12pt; color: #000000;">Α. Να αποδειχτεί ότι</span></p>
<p><span style="font-size: 12pt; color: #000000;"> 1. <img class="ee_img tr_noresize" alt="\alpha - 2\beta = 1" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%20-%202%5Cbeta%20%20%3D%201" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyOeI0IaaGOmaiabek7aIjabg2da9iaaigdaaaa!3C1E!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 12pt; color: #000000;"> 2. <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to = \left( {\alpha \,,\,\alpha } \right)\,\,\,\kappa \alpha \iota \,\,\, \beta \limits^ \to = \left( {\alpha - 2\,,\,2 - \alpha } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%5C%2C%2C%5C%2C%5Calpha%20%7D%20%5Cright%29%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%20-%202%5C%2C%2C%5C%2C2%20-%20%5Calpha%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiabeg7aHjaaykW7caGGSaGaaGPaVlabeg7aHbGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVJqaaKqzagGaa8NUdiaa-f7acaWF5oGccaaMc8UaaGPaVlaaykW7daWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakiabg2da9maabmaabaGaeqySdeMaeyOeI0IaaGOmaiaaykW7caGGSaGaaGPaVlaaikdacqGHsislcqaHXoqyaiaawIcacaGLPaaaaaa!61A0!" style="vertical-align: middle;" /> <span style="position: relative; top: 7pt;"> </span></span></p>
<p><span style="font-size: 12pt; color: #000000;">Β. Αν επιπλέον ισχύει <img class="ee_img tr_noresize" alt="\left( { \alpha \limits^ \to \, ,\limits^{^{ \,\limits^ \wedge }} \, \alpha \limits^ \to \, - \beta \limits^ \to } \right) = {45^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20%2C%5Climits%5E%7B%5E%7B%20%5C%2C%5Climits%5E%20%5Cwedge%20%20%7D%7D%20%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20-%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%20%3D%20%7B45%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiaaykW7daWfGaqaaiaacYcaaSqabeaadaahaaadbeqaamaaxacabaGaaGPaVdqabeaajugqbiabgEIizdaaaaaaaOGaaGPaVpaaxacabaGaeqySdegaleqabaGaeyOKH4kaaOGaaGPaVlabgkHiTmaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaGccaGLOaGaayzkaaGaeyypa0JaaGinaiaaiwdadaahaaWcbeqaaGGaaiab-97aVbaaaaa!51F1!" style="vertical-align: middle;" /><span style="position: relative; top: 12pt;"></span> να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\alpha = 1" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%20%3D%201" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyypa0JaaGymaaaa!38D4!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 12pt; color: #000000;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to = \left( {\alpha \,,\,2\beta + 1} \right),\,\,\, \beta \limits^ \to = \left( {4\beta - \alpha \,,\,1 - 2\beta } \right),\,\,\,\alpha ,\beta \in R" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%5C%2C%2C%5C%2C2%5Cbeta%20%20%2B%201%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B4%5Cbeta%20%20-%20%5Calpha%20%5C%2C%2C%5C%2C1%20-%202%5Cbeta%20%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Calpha%20%2C%5Cbeta%20%20%5Cin%20R" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiabeg7aHjaaykW7caGGSaGaaGPaVlaaikdacqaHYoGycqGHRaWkcaaIXaaacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVpaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaaI0aGaeqOSdiMaeyOeI0IaeqySdeMaaGPaVlaacYcacaaMc8UaaGymaiabgkHiTiaaikdacqaHYoGyaiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8UaeqySdeMaaiilaiabek7aIjabgIGiolaadkfaaaa!6954!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 12pt; color: #000000;">Αν <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to \bot \beta \limits^ \to \,\,\,\kappa \alpha \iota \,\,\,\alpha > 2\beta " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%5Cbot%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%5Calpha%20%20%3E%202%5Cbeta%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHLkIxdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakiaaykW7caaMc8UaaGPaVJqaaKqzagGaa8NUdiaa-f7acaWF5oGccaaMc8UaaGPaVlaaykW7cqaHXoqycqGH+aGpcaaIYaGaeqOSdigaaa!51DD!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 12pt; color: #000000;">Α. Να αποδειχτεί ότι</span></p>
<p><span style="font-size: 12pt; color: #000000;"> 1. <img class="ee_img tr_noresize" alt="\alpha - 2\beta = 1" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%20-%202%5Cbeta%20%20%3D%201" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyOeI0IaaGOmaiabek7aIjabg2da9iaaigdaaaa!3C1E!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 12pt; color: #000000;"> 2. <img class="ee_img tr_noresize" alt=" \alpha \limits^ \to = \left( {\alpha \,,\,\alpha } \right)\,\,\,\kappa \alpha \iota \,\,\, \beta \limits^ \to = \left( {\alpha - 2\,,\,2 - \alpha } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%5C%2C%2C%5C%2C%5Calpha%20%7D%20%5Cright%29%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%20-%202%5C%2C%2C%5C%2C2%20-%20%5Calpha%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiabeg7aHjaaykW7caGGSaGaaGPaVlabeg7aHbGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVJqaaKqzagGaa8NUdiaa-f7acaWF5oGccaaMc8UaaGPaVlaaykW7daWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakiabg2da9maabmaabaGaeqySdeMaeyOeI0IaaGOmaiaaykW7caGGSaGaaGPaVlaaikdacqGHsislcqaHXoqyaiaawIcacaGLPaaaaaa!61A0!" style="vertical-align: middle;" /> <span style="position: relative; top: 7pt;"> </span></span></p>
<p><span style="font-size: 12pt; color: #000000;">Β. Αν επιπλέον ισχύει <img class="ee_img tr_noresize" alt="\left( { \alpha \limits^ \to \, ,\limits^{^{ \,\limits^ \wedge }} \, \alpha \limits^ \to \, - \beta \limits^ \to } \right) = {45^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20%2C%5Climits%5E%7B%5E%7B%20%5C%2C%5Climits%5E%20%5Cwedge%20%20%7D%7D%20%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%20-%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%20%3D%20%7B45%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiaaykW7daWfGaqaaiaacYcaaSqabeaadaahaaadbeqaamaaxacabaGaaGPaVdqabeaajugqbiabgEIizdaaaaaaaOGaaGPaVpaaxacabaGaeqySdegaleqabaGaeyOKH4kaaOGaaGPaVlabgkHiTmaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaGccaGLOaGaayzkaaGaeyypa0JaaGinaiaaiwdadaahaaWcbeqaaGGaaiab-97aVbaaaaa!51F1!" style="vertical-align: middle;" /><span style="position: relative; top: 12pt;"></span> να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\alpha = 1" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%20%3D%201" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyypa0JaaGymaaaa!38D4!" style="vertical-align: middle;" />.</span></p>
17. Β΄Λυκείου/ Κατεύθυνση/ Συντεταγμένες διανύσματος
2015-11-11T13:17:49+02:00
2015-11-11T13:17:49+02:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/179-17-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CF%83%CF%85%CE%BD%CF%84%CE%B5%CF%84%CE%B1%CE%B3%CE%BC%CE%AD%CE%BD%CE%B5%CF%82-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%BF%CF%82
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 14pt; color: #000000;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" alt=" u\limits^ \to = \left( {4\beta + \gamma - \alpha \,\,,\,\,\alpha \gamma + 4{\beta ^2}} \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20u%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B4%5Cbeta%20%20%2B%20%5Cgamma%20%20-%20%5Calpha%20%5C%2C%5C%2C%2C%5C%2C%5C%2C%5Calpha%20%5Cgamma%20%20%2B%204%7B%5Cbeta%20%5E2%7D%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaaI0aGaeqOSdiMaey4kaSIaeq4SdCMaeyOeI0IaeqySdeMaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7cqaHXoqycqaHZoWzcqGHRaWkcaaI0aGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaaa!5106!" style="vertical-align: middle;" /> </span></p>
<p><span style="font-size: 14pt; color: #000000;">και <img class="ee_img tr_noresize" alt=" v\limits^ \to = \left( {\alpha - 2\beta ,0} \right),\,\,\,\alpha ,\beta ,\gamma \in R" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20v%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%20-%202%5Cbeta%20%2C0%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Calpha%20%2C%5Cbeta%20%2C%5Cgamma%20%20%5Cin%20R" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG2baaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacqaHXoqycqGHsislcaaIYaGaeqOSdiMaaiilaiaaicdaaiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8UaeqySdeMaaiilaiabek7aIjaacYcacqaHZoWzcqGHiiIZcaWGsbaaaa!4F83!" style="vertical-align: middle;" /> <span style="position: relative; top: 11pt;"></span>.</span></p>
<p><span style="font-size: 14pt; color: #000000;">Αν <img class="ee_img tr_noresize" alt=" u\limits^ \to ||y'y" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20u%5Climits%5E%20%5Cto%20%20%7C%7Cy%27y" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaaiiFaiaacYhaceWG5bGbauaacaWG5baaaa!3CB5!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 14pt; color: #000000;">Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\left( {\widehat {{\rm O}x, u\limits^ \to }} \right) = {90^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cwidehat%20%7B%7B%5Crm%20O%7Dx%2C%20u%5Climits%5E%20%5Cto%20%20%7D%7D%20%5Cright%29%20%3D%20%7B90%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaqiaaqaaiabf+5apjaadIhacaGGSaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaaGccaGLcmaaaiaawIcacaGLPaaacqGH9aqpcaaI5aGaaGimamaaCaaaleqabaaccaGae83Vd8gaaaaa!4290!" style="vertical-align: middle;" />.</span><br /><span style="font-size: 14pt; color: #000000;"> Β. Δίνεται ότι <img class="ee_img tr_noresize" alt="\left| { u\limits^ \to } \right| = \left| { v\limits^ \to } \right|\," src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20u%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20v%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%5C%2C" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiaadwhaaSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacqGH9aqpdaabdaqaamaaxacabaGaamODaaWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiaaykW7aaa!44BD!" style="vertical-align: middle;" /> <span style="position: relative; top: 9pt;"> </span> και <img class="ee_img tr_noresize" alt="\left( {\widehat {{\rm O}x,\, v\limits^ \to }} \right) = {0^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cwidehat%20%7B%7B%5Crm%20O%7Dx%2C%5C%2C%20v%5Climits%5E%20%5Cto%20%20%7D%7D%20%5Cright%29%20%3D%20%7B0%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaqiaaqaaiabf+5apjaadIhacaGGSaGaaGPaVpaaxacabaGaamODaaWcbeqaaiabgkziUcaaaOGaayPadaaacaGLOaGaayzkaaGaeyypa0JaaGimamaaCaaaleqabaaccaGae83Vd8gaaaaa!4359!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 14pt; color: #000000;"> 1. Να βρεθούν τα διανύσματα <img class="ee_img tr_noresize" alt=" u\limits^ \to ,\,\,\, v\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20u%5Climits%5E%20%5Cto%20%20%2C%5C%2C%5C%2C%5C%2C%20v%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaaiilaiaaykW7caaMc8UaaGPaVpaaxacabaGaamODaaWcbeqaaiabgkziUcaaaaa!4130!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 14pt; color: #000000;"> 2. Αν επιπλέον <img class="ee_img tr_noresize" alt="\left( {\alpha \, u\limits^ \to + v\limits^ \to } \right)||\left( { u\limits^ \to + \gamma \, v\limits^ \to } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Calpha%20%5C%2C%20u%5Climits%5E%20%5Cto%20%20%20%2B%20%20v%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%7C%7C%5Cleft%28%20%7B%20u%5Climits%5E%20%5Cto%20%20%20%2B%20%5Cgamma%20%5C%2C%20v%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqaHXoqycaaMc8+aaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaey4kaSYaaCbiaeaacaWG2baaleqabaGaeyOKH4kaaaGccaGLOaGaayzkaaGaaiiFaiaacYhadaqadaqaamaaxacabaGaamyDaaWcbeqaaiabgkziUcaakiabgUcaRiabeo7aNjaaykW7daWfGaqaaiaadAhaaSqabeaacqGHsgIRaaaakiaawIcacaGLPaaaaaa!4F90!" style="vertical-align: middle;" /><span style="position: relative; top: 9pt;"> </span> να υπολογιστούν οι αριθμοί <img class="ee_img tr_noresize" alt="\alpha ,\beta ,\gamma " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%2C%5Cbeta%20%2C%5Cgamma%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIjaacYcacqaHZoWzaaa!3BBB!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 14pt; color: #000000;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" alt=" u\limits^ \to = \left( {4\beta + \gamma - \alpha \,\,,\,\,\alpha \gamma + 4{\beta ^2}} \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20u%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B4%5Cbeta%20%20%2B%20%5Cgamma%20%20-%20%5Calpha%20%5C%2C%5C%2C%2C%5C%2C%5C%2C%5Calpha%20%5Cgamma%20%20%2B%204%7B%5Cbeta%20%5E2%7D%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaaI0aGaeqOSdiMaey4kaSIaeq4SdCMaeyOeI0IaeqySdeMaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7cqaHXoqycqaHZoWzcqGHRaWkcaaI0aGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaaa!5106!" style="vertical-align: middle;" /> </span></p>
<p><span style="font-size: 14pt; color: #000000;">και <img class="ee_img tr_noresize" alt=" v\limits^ \to = \left( {\alpha - 2\beta ,0} \right),\,\,\,\alpha ,\beta ,\gamma \in R" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20v%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Calpha%20%20-%202%5Cbeta%20%2C0%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Calpha%20%2C%5Cbeta%20%2C%5Cgamma%20%20%5Cin%20R" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG2baaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacqaHXoqycqGHsislcaaIYaGaeqOSdiMaaiilaiaaicdaaiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8UaeqySdeMaaiilaiabek7aIjaacYcacqaHZoWzcqGHiiIZcaWGsbaaaa!4F83!" style="vertical-align: middle;" /> <span style="position: relative; top: 11pt;"></span>.</span></p>
<p><span style="font-size: 14pt; color: #000000;">Αν <img class="ee_img tr_noresize" alt=" u\limits^ \to ||y'y" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20u%5Climits%5E%20%5Cto%20%20%7C%7Cy%27y" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaaiiFaiaacYhaceWG5bGbauaacaWG5baaaa!3CB5!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 14pt; color: #000000;">Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" alt="\left( {\widehat {{\rm O}x, u\limits^ \to }} \right) = {90^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cwidehat%20%7B%7B%5Crm%20O%7Dx%2C%20u%5Climits%5E%20%5Cto%20%20%7D%7D%20%5Cright%29%20%3D%20%7B90%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaqiaaqaaiabf+5apjaadIhacaGGSaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaaGccaGLcmaaaiaawIcacaGLPaaacqGH9aqpcaaI5aGaaGimamaaCaaaleqabaaccaGae83Vd8gaaaaa!4290!" style="vertical-align: middle;" />.</span><br /><span style="font-size: 14pt; color: #000000;"> Β. Δίνεται ότι <img class="ee_img tr_noresize" alt="\left| { u\limits^ \to } \right| = \left| { v\limits^ \to } \right|\," src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20u%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20v%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%5C%2C" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiaadwhaaSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacqGH9aqpdaabdaqaamaaxacabaGaamODaaWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiaaykW7aaa!44BD!" style="vertical-align: middle;" /> <span style="position: relative; top: 9pt;"> </span> και <img class="ee_img tr_noresize" alt="\left( {\widehat {{\rm O}x,\, v\limits^ \to }} \right) = {0^o}" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Cwidehat%20%7B%7B%5Crm%20O%7Dx%2C%5C%2C%20v%5Climits%5E%20%5Cto%20%20%7D%7D%20%5Cright%29%20%3D%20%7B0%5Eo%7D" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaqiaaqaaiabf+5apjaadIhacaGGSaGaaGPaVpaaxacabaGaamODaaWcbeqaaiabgkziUcaaaOGaayPadaaacaGLOaGaayzkaaGaeyypa0JaaGimamaaCaaaleqabaaccaGae83Vd8gaaaaa!4359!" style="vertical-align: middle;" /></span></p>
<p><span style="font-size: 14pt; color: #000000;"> 1. Να βρεθούν τα διανύσματα <img class="ee_img tr_noresize" alt=" u\limits^ \to ,\,\,\, v\limits^ \to " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20u%5Climits%5E%20%5Cto%20%20%2C%5C%2C%5C%2C%5C%2C%20v%5Climits%5E%20%5Cto%20%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaaiilaiaaykW7caaMc8UaaGPaVpaaxacabaGaamODaaWcbeqaaiabgkziUcaaaaa!4130!" style="vertical-align: middle;" />.</span></p>
<p><span style="font-size: 14pt; color: #000000;"> 2. Αν επιπλέον <img class="ee_img tr_noresize" alt="\left( {\alpha \, u\limits^ \to + v\limits^ \to } \right)||\left( { u\limits^ \to + \gamma \, v\limits^ \to } \right)" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%5Calpha%20%5C%2C%20u%5Climits%5E%20%5Cto%20%20%20%2B%20%20v%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29%7C%7C%5Cleft%28%20%7B%20u%5Climits%5E%20%5Cto%20%20%20%2B%20%5Cgamma%20%5C%2C%20v%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacqaHXoqycaaMc8+aaCbiaeaacaWG1baaleqabaGaeyOKH4kaaOGaey4kaSYaaCbiaeaacaWG2baaleqabaGaeyOKH4kaaaGccaGLOaGaayzkaaGaaiiFaiaacYhadaqadaqaamaaxacabaGaamyDaaWcbeqaaiabgkziUcaakiabgUcaRiabeo7aNjaaykW7daWfGaqaaiaadAhaaSqabeaacqGHsgIRaaaakiaawIcacaGLPaaaaaa!4F90!" style="vertical-align: middle;" /><span style="position: relative; top: 9pt;"> </span> να υπολογιστούν οι αριθμοί <img class="ee_img tr_noresize" alt="\alpha ,\beta ,\gamma " src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20%2C%5Cbeta%20%2C%5Cgamma%20" eeimg="1" longdesc="http://docs.google.com/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0dg9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIjaacYcacqaHZoWzaaa!3BBB!" style="vertical-align: middle;" /></span></p>
16. Β΄Λυκείου/ Κατεύθυνση/ Εσωτερικό γινόμενο διανυσμάτων
2015-06-11T18:30:04+03:00
2015-06-11T18:30:04+03:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/171-16-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt;">Δίνονται τα διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7Bx%2Cy%7D%20%5Cright%29%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7Bx%20-%20y%2Cx%20%2B%20y%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2Cx%2Cy%20%5Cin%20R" alt=" alpha limits^ to = left( {x,y} right),,,kappa alpha iota ,,, beta limits^ to = left( {x - y,x + y} right),,,,x,y in R" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVJGaaiab-P7aRjab-f7aHjab-L7aPjaaykW7caaMc8UaaGPaVpaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaWG4bGaeyOeI0IaamyEaiaacYcacaWG4bGaey4kaSIaamyEaaGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7caWG4bGaaiilaiaadMhacqGHiiIZcaWGsbaaaa!63EB!" />.</span></p>
<p><span style="font-size: 12pt;">Αν ισχύει <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%201" alt=" alpha limits^ to beta limits^ to = 1" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGcdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakiabg2da9iaaigdaaaa!3EDA!" /> </span></p>
<p><span style="font-size: 12pt;">Α. Να δειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to } right| = left| { alpha limits^ to - beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHsisldaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aaaa!492F!" />.</span> </p>
<p><span style="font-size: 12pt;">Β. Να υπολογιστούν οι γωνίες <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29" alt="left( { alpha limits^ to ,,,,, beta limits^ to } right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiaaykW7caaMc8UaaiilaiaaykW7caaMc8+aaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaaakiaawIcacaGLPaaaaaa!457D!" /> και <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5C%2C%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%2C%5C%2C%5C%2C%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29" alt=",left( { alpha limits^ to ,,,,,,, alpha limits^ to - beta limits^ to } right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVpaabmaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7caaMc8+aaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHsisldaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaayjkaiaawMcaaaaa!4EEA!" />. </span> </p>
<p><span style="font-size: 12pt;">Γ. Αν η γωνία που σχηματίζει το διάνυσμα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaaaaa!392E!" /> με τον ημιάξονα Οx είναι 135<sup>ο</sup> να υπολογιστούν</span> </p>
<p><span style="font-size: 12pt;"> οι συντεταγμένες των διανυσμάτων <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to ,,kappa alpha iota ,,, beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVJGaaiab-P7aRjab-f7aHjab-L7aPjaaykW7caaMc8UaaGPaVpaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaaa!49C1!" /></span></p>
<p> </p>
<p><span style="font-size: 12pt;">Δίνονται τα διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7Bx%2Cy%7D%20%5Cright%29%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7Bx%20-%20y%2Cx%20%2B%20y%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2Cx%2Cy%20%5Cin%20R" alt=" alpha limits^ to = left( {x,y} right),,,kappa alpha iota ,,, beta limits^ to = left( {x - y,x + y} right),,,,x,y in R" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVJGaaiab-P7aRjab-f7aHjab-L7aPjaaykW7caaMc8UaaGPaVpaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaWG4bGaeyOeI0IaamyEaiaacYcacaWG4bGaey4kaSIaamyEaaGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7caWG4bGaaiilaiaadMhacqGHiiIZcaWGsbaaaa!63EB!" />.</span></p>
<p><span style="font-size: 12pt;">Αν ισχύει <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%201" alt=" alpha limits^ to beta limits^ to = 1" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGcdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakiabg2da9iaaigdaaaa!3EDA!" /> </span></p>
<p><span style="font-size: 12pt;">Α. Να δειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to } right| = left| { alpha limits^ to - beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHsisldaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aaaa!492F!" />.</span> </p>
<p><span style="font-size: 12pt;">Β. Να υπολογιστούν οι γωνίες <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29" alt="left( { alpha limits^ to ,,,,, beta limits^ to } right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiaaykW7caaMc8UaaiilaiaaykW7caaMc8+aaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaaakiaawIcacaGLPaaaaaa!457D!" /> και <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5C%2C%5Cleft%28%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%2C%5C%2C%5C%2C%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%29" alt=",left( { alpha limits^ to ,,,,,,, alpha limits^ to - beta limits^ to } right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVpaabmaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7caaMc8+aaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHsisldaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaayjkaiaawMcaaaaa!4EEA!" />. </span> </p>
<p><span style="font-size: 12pt;">Γ. Αν η γωνία που σχηματίζει το διάνυσμα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaaaaa!392E!" /> με τον ημιάξονα Οx είναι 135<sup>ο</sup> να υπολογιστούν</span> </p>
<p><span style="font-size: 12pt;"> οι συντεταγμένες των διανυσμάτων <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to ,,kappa alpha iota ,,, beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVJGaaiab-P7aRjab-f7aHjab-L7aPjaaykW7caaMc8UaaGPaVpaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaaa!49C1!" /></span></p>
<p> </p>
15. Β΄Λυκείου/ Κατεύθυνση/ Συντεταγμένες διανύσματος
2015-06-11T18:20:12+03:00
2015-06-11T18:20:12+03:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/170-15-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt;">Δίνονται τα διανύσματα</span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Ckappa%20%20-%20%5Clambda%20%5C%2C%2C%5C%2C%5Clambda%20%20-%20%5Ckappa%20%20%2B%201%7D%20%5Cright%29%5C%2C" alt=" alpha limits^ to = left( {kappa - lambda ,,,lambda - kappa + 1} right)," longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiabeQ7aRjabgkHiTiabeU7aSjaaykW7caGGSaGaaGPaVlabeU7aSjabgkHiTiabeQ7aRjabgUcaRiaaigdaaiaawIcacaGLPaaacaaMc8oaaa!4B5A!" /> και <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B2%5Ckappa%20%20%2B%20%5Clambda%20%5C%2C%2C%5C%2C%5Clambda%20%20%2B%202%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Ckappa%20%2C%5Clambda%20%20%5Cin%20R" alt=" beta limits^ to = left( {2kappa + lambda ,,,lambda + 2} right),,,,kappa ,lambda in R" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiaaikdacqaH6oWAcqGHRaWkcqaH7oaBcaaMc8UaaiilaiaaykW7cqaH7oaBcqGHRaWkcaaIYaaacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlabeQ7aRjaacYcacqaH7oaBcqGHiiIZcaWGsbaaaa!53A6!" />. </span></p>
<p><span style="font-size: 12pt;">Αν <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%201" alt="left| { alpha limits^ to } right| = 1" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9iaaigdaaaa!3E3C!" /> </span></p>
<p><span style="font-size: 12pt;">Α. Να αποδείξετε ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B0%2C1%7D%20%5Cright%29%5C%2C" alt=" alpha limits^ to = left( {0,1} right)," longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiaaicdacaGGSaGaaGymaaGaayjkaiaawMcaaiaaykW7aaa!3F76!" /> ή <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B1%2C0%7D%20%5Cright%29" alt=", alpha limits^ to = left( {1,0} right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGPaVpaaxacabaGaeqySdegaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaaIXaGaaiilaiaaicdaaiaawIcacaGLPaaaaaa!3F76!" />. </span> </p>
<p><span style="font-size: 12pt;">Β. Αν επιπλέον <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%2F%2F%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20" alt=" beta limits^ to //, alpha limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaGccaGGVaGaai4laiaaykW7daWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaaa!4021!" /> να βρεθεί το διάνυσμα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaaaaa!3951!" />.</span></p>
<p><span style="font-size: 12pt;">Δίνονται τα διανύσματα</span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B%5Ckappa%20%20-%20%5Clambda%20%5C%2C%2C%5C%2C%5Clambda%20%20-%20%5Ckappa%20%20%2B%201%7D%20%5Cright%29%5C%2C" alt=" alpha limits^ to = left( {kappa - lambda ,,,lambda - kappa + 1} right)," longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiabeQ7aRjabgkHiTiabeU7aSjaaykW7caGGSaGaaGPaVlabeU7aSjabgkHiTiabeQ7aRjabgUcaRiaaigdaaiaawIcacaGLPaaacaaMc8oaaa!4B5A!" /> και <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B2%5Ckappa%20%20%2B%20%5Clambda%20%5C%2C%2C%5C%2C%5Clambda%20%20%2B%202%7D%20%5Cright%29%2C%5C%2C%5C%2C%5C%2C%5Ckappa%20%2C%5Clambda%20%20%5Cin%20R" alt=" beta limits^ to = left( {2kappa + lambda ,,,lambda + 2} right),,,,kappa ,lambda in R" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiaaikdacqaH6oWAcqGHRaWkcqaH7oaBcaaMc8UaaiilaiaaykW7cqaH7oaBcqGHRaWkcaaIYaaacaGLOaGaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlabeQ7aRjaacYcacqaH7oaBcqGHiiIZcaWGsbaaaa!53A6!" />. </span></p>
<p><span style="font-size: 12pt;">Αν <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%201" alt="left| { alpha limits^ to } right| = 1" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9iaaigdaaaa!3E3C!" /> </span></p>
<p><span style="font-size: 12pt;">Α. Να αποδείξετε ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B0%2C1%7D%20%5Cright%29%5C%2C" alt=" alpha limits^ to = left( {0,1} right)," longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpdaqadaqaaiaaicdacaGGSaGaaGymaaGaayjkaiaawMcaaiaaykW7aaa!3F76!" /> ή <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Cleft%28%20%7B1%2C0%7D%20%5Cright%29" alt=", alpha limits^ to = left( {1,0} right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaGPaVpaaxacabaGaeqySdegaleqabaGaeyOKH4kaaOGaeyypa0ZaaeWaaeaacaaIXaGaaiilaiaaicdaaiaawIcacaGLPaaaaaa!3F76!" />. </span> </p>
<p><span style="font-size: 12pt;">Β. Αν επιπλέον <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%2F%2F%5C%2C%20%5Calpha%20%5Climits%5E%20%5Cto%20%20" alt=" beta limits^ to //, alpha limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaGccaGGVaGaai4laiaaykW7daWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaaa!4021!" /> να βρεθεί το διάνυσμα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaaaaa!3951!" />.</span></p>
14. Β΄Λυκείου/ Κατεύθυνση/ Εσωτερικό γινόμενο διανυσμάτων
2015-06-11T18:12:03+03:00
2015-06-11T18:12:03+03:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/169-14-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to ,,,kappa alpha iota ,,, beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugWbiaa-P7acaWFXoGaa8xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaaaaa!4B43!" /> για τα οποία ισχύει ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20-%202%5Cleft%7C%20%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to - 2 beta limits^ to } right| = left| { alpha limits^ to } right| - 2left| { beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiabgkHiTiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacqGHsislcaaIYaWaaqWaaeaadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aaaa!5297!" /></span></p>
<p><span style="font-size: 12pt;">Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%5Cuparrow%20%20%5Cuparrow%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to uparrow uparrow beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHrgsRcqGHrgsRdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaaa!40E5!" />.</span> </p>
<p><span style="font-size: 12pt;">Β. Αν <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Clambda%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to = lambda beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpcqaH7oaBdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaaa!3FC9!" /> να αποδειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Clambda%20%20%5Cge%202" alt="lambda ge 2" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyyzImRaaGOmaaaa!398F!" />.</span> </p>
<p><span style="font-size: 12pt;">Γ. Να αποδειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%2B%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%2B%202%5Cleft%7C%20%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to + 2 beta limits^ to } right| = left| { alpha limits^ to } right| + 2left| { beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiabgUcaRiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacqGHRaWkcaaIYaWaaqWaaeaadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aaaa!5281!" />.</span> </p>
<p><span style="font-size: 12pt;">Δ. Αν οι αριθμοί <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%2B%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to - 2 beta limits^ to } right|,,,kappa alpha iota ,,,left| { alpha limits^ to + 2 beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiabgkHiTiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiaaykW7caaMc8UaaGPaVJqaaKqzahGaa8NUdiaa-f7acaWF5oGccaaMc8UaaGPaVlaaykW7daabdaqaamaaxacabaGaeqySdegaleqabaGaeyOKH4kaaOGaey4kaSIaaGOmamaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaGccaGLhWUaayjcSdaaaa!5C98!" /> είναι λύσεις της εξίσωσης <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bx%5E2%7D%20-%20%5Cleft%28%20%7B%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%2B%201%7D%20%5Cright%29x%20%2B%2012%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%5E2%7D%20%3D%200" alt="{x^2} - left( {left| { alpha limits^ to } right| + 1} right)x + 12{ beta limits^ to ^2} = 0" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTmaabmaabaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabgUcaRiaaigdaaiaawIcacaGLPaaacaWG4bGaey4kaSIaaGymaiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakmaaCaaaleqabaGaaGOmaaaakiabg2da9iaaicdaaaa!4C47!" /></span></p>
<p><span style="font-size: 12pt;"> να υπολογιστούν : <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%2C%5C%2C%5C%2C%5C%2C%5Cleft%7C%20%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to } right|,,,,left| { beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiaacYcacaaMc8UaaGPaVlaaykW7daabdaqaamaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaGccaGLhWUaayjcSdaaaa!48AE!" /></span>.</p>
<p><span style="font-size: 12pt;">Δίνονται τα μη μηδενικά διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to ,,,kappa alpha iota ,,, beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugWbiaa-P7acaWFXoGaa8xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqaHYoGyaSqabeaacqGHsgIRaaaaaa!4B43!" /> για τα οποία ισχύει ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20-%202%5Cleft%7C%20%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to - 2 beta limits^ to } right| = left| { alpha limits^ to } right| - 2left| { beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiabgkHiTiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacqGHsislcaaIYaWaaqWaaeaadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aaaa!5297!" /></span></p>
<p><span style="font-size: 12pt;">Α. Να αποδειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%5Cuparrow%20%20%5Cuparrow%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to uparrow uparrow beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGHrgsRcqGHrgsRdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaaa!40E5!" />.</span> </p>
<p><span style="font-size: 12pt;">Β. Αν <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%3D%20%5Clambda%20%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20" alt=" alpha limits^ to = lambda beta limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaGccqGH9aqpcqaH7oaBdaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaaa!3FC9!" /> να αποδειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Clambda%20%20%5Cge%202" alt="lambda ge 2" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyyzImRaaGOmaaaa!398F!" />.</span> </p>
<p><span style="font-size: 12pt;">Γ. Να αποδειχτεί ότι <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%2B%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%3D%20%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%2B%202%5Cleft%7C%20%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to + 2 beta limits^ to } right| = left| { alpha limits^ to } right| + 2left| { beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiabgUcaRiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabg2da9maaemaabaWaaCbiaeaacqaHXoqyaSqabeaacqGHsgIRaaaakiaawEa7caGLiWoacqGHRaWkcaaIYaWaaqWaaeaadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aaaa!5281!" />.</span> </p>
<p><span style="font-size: 12pt;">Δ. Αν οι αριθμοί <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20-%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%20%2B%202%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to - 2 beta limits^ to } right|,,,kappa alpha iota ,,,left| { alpha limits^ to + 2 beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaakiabgkHiTiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiaaykW7caaMc8UaaGPaVJqaaKqzahGaa8NUdiaa-f7acaWF5oGccaaMc8UaaGPaVlaaykW7daabdaqaamaaxacabaGaeqySdegaleqabaGaeyOKH4kaaOGaey4kaSIaaGOmamaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaGccaGLhWUaayjcSdaaaa!5C98!" /> είναι λύσεις της εξίσωσης <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bx%5E2%7D%20-%20%5Cleft%28%20%7B%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%20%2B%201%7D%20%5Cright%29x%20%2B%2012%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%5E2%7D%20%3D%200" alt="{x^2} - left( {left| { alpha limits^ to } right| + 1} right)x + 12{ beta limits^ to ^2} = 0" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTmaabmaabaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiabgUcaRiaaigdaaiaawIcacaGLPaaacaWG4bGaey4kaSIaaGymaiaaikdadaWfGaqaaiabek7aIbWcbeqaaiabgkziUcaakmaaCaaaleqabaGaaGOmaaaakiabg2da9iaaicdaaaa!4C47!" /></span></p>
<p><span style="font-size: 12pt;"> να υπολογιστούν : <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%7B%20%5Calpha%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C%2C%5C%2C%5C%2C%5C%2C%5Cleft%7C%20%7B%20%5Cbeta%20%5Climits%5E%20%5Cto%20%20%7D%20%5Cright%7C" alt="left| { alpha limits^ to } right|,,,,left| { beta limits^ to } right|" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVC0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaqWaaeaadaWfGaqaaiabeg7aHbWcbeqaaiabgkziUcaaaOGaay5bSlaawIa7aiaacYcacaaMc8UaaGPaVlaaykW7daabdaqaamaaxacabaGaeqOSdigaleqabaGaeyOKH4kaaaGccaGLhWUaayjcSdaaaa!48AE!" /></span>.</p>
13. Β΄Λυκείου/ Κατεύθυνση/ Διανύσματα
2014-10-09T19:07:04+03:00
2014-10-09T19:07:04+03:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/142-13-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt;">Δίνεται τρίγωνο ΑΒΓ και τα σημεία Δ, Ε, Μ ώστε</span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=2%20%7B%5CGamma%20%5CDelta%20%7D%5Climits%5E%20%5Cto%20%20%20%3D%20%20%7B%5CGamma%20%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20%20%2B%20%20%7B%5CGamma%20%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20" alt="2 {Gamma Delta }limits^ to = {Gamma {rm A}}limits^ to + {Gamma {rm B}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaxacabaGaeu4KdCKaeuiLdqealeqabaGaeyOKH4kaaOGaeyypa0ZaaCbiaeaacqqHtoWrcqqHroqqaSqabeaacqGHsgIRaaGccqGHRaWkdaWfGaqaaiabfo5ahjabfk5acbWcbeqaaiabgkziUcaaaaa!47B4!" /></span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=3%20%7B%7B%5Crm%20E%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20%20%3D%20%20%7B%7B%5Crm%20A%7D%7B%5Crm%20E%7D%7D%5Climits%5E%20%5Cto%20%20" alt="3 {{rm E}Gamma }limits^ to = {{rm A}{rm E}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaxacabaGaeuyLduKaeu4KdCealeqabaGaeyOKH4kaaOGaeyypa0ZaaCbiaeaacqqHroqqcqqHvoqraSqabeaacqGHsgIRaaaaaa!41CB!" /></span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=2%20%7B%7B%5Crm%20B%7D%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20%20%3D%203%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" alt="2 {{rm B}{rm M}}limits^ to = 3 {{rm B}Gamma }limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaxacabaGaeuOKdiKaeuiNd0ealeqabaGaeyOKH4kaaOGaeyypa0JaaG4mamaaxacabaGaeuOKdiKaeu4KdCealeqabaGaeyOKH4kaaaaa!4292!" /> </span></p>
<p><span style="font-size: 12pt;">Α. Να εκφραστούν τα διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%5CDelta%20%7D%5Climits%5E%20%5Cto%20%20%2C%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20E%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm A}Delta }limits^ to ,,,, {{rm A}{rm E}}limits^ to ,,,kappa alpha iota ,,, {{rm A}{rm M}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHuoaraSqabeaacqGHsgIRaaGccaGGSaGaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHroqqcqqHvoqraSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7iiaacqWF6oWAcqWFXoqycqWF5oqAcaaMc8UaaGPaVlaaykW7daWfGaqaaiabfg5abjabfY5anbWcbeqaaiabgkziUcaaaaa!58A2!" /> ως προς <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%5C%2C%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm A}{rm B},}limits^ to ,,,kappa alpha iota ,,, {{rm A}Gamma }limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqcaaMc8oaleqabaGaeyOKH4kaaOGaaGPaVlaaykW7caaMc8occaGae8NUdSMae8xSdeMae8xUdKMaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHroqqcqqHtoWraSqabeaacqGHsgIRaaaaaa!4FC2!" /></span></p>
<p><span style="font-size: 12pt;">Β. Να εκφραστούν τα διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%5CDelta%20%7B%5Crm%20E%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%5CDelta%20%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20" alt=" {Delta {rm E}}limits^ to ,,,kappa alpha iota ,,, {Delta {rm M}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHuoarcqqHvoqraSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugWbiaa-P7acaWFXoGaa8xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHuoarcqqHCoqtaSqabeaacqGHsgIRaaaaaa!4E4F!" /> ως προς <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%5C%2C%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm A}{rm B},}limits^ to ,,,kappa alpha iota ,,, {{rm A}Gamma }limits^ to " longdesc="http://ematha.vassiliadis.edu.gr/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqcaaMc8oaleqabaGaeyOKH4kaaOGaaGPaVlaaykW7caaMc8occaGae8NUdSMae8xSdeMae8xUdKMaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHroqqcqqHtoWraSqabeaacqGHsgIRaaaaaa!4FC2!" /></span></p>
<p><span style="font-size: 12pt;">Γ. Να αποδειχτεί ότι το σημείο Ε είναι το μέσο του ΔΜ.</span></p>
<p><span style="font-size: 12pt;">Δίνεται τρίγωνο ΑΒΓ και τα σημεία Δ, Ε, Μ ώστε</span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=2%20%7B%5CGamma%20%5CDelta%20%7D%5Climits%5E%20%5Cto%20%20%20%3D%20%20%7B%5CGamma%20%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20%20%2B%20%20%7B%5CGamma%20%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20" alt="2 {Gamma Delta }limits^ to = {Gamma {rm A}}limits^ to + {Gamma {rm B}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaxacabaGaeu4KdCKaeuiLdqealeqabaGaeyOKH4kaaOGaeyypa0ZaaCbiaeaacqqHtoWrcqqHroqqaSqabeaacqGHsgIRaaGccqGHRaWkdaWfGaqaaiabfo5ahjabfk5acbWcbeqaaiabgkziUcaaaaa!47B4!" /></span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=3%20%7B%7B%5Crm%20E%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20%20%3D%20%20%7B%7B%5Crm%20A%7D%7B%5Crm%20E%7D%7D%5Climits%5E%20%5Cto%20%20" alt="3 {{rm E}Gamma }limits^ to = {{rm A}{rm E}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaxacabaGaeuyLduKaeu4KdCealeqabaGaeyOKH4kaaOGaeyypa0ZaaCbiaeaacqqHroqqcqqHvoqraSqabeaacqGHsgIRaaaaaa!41CB!" /></span></p>
<p><span style="font-size: 12pt;"> <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=2%20%7B%7B%5Crm%20B%7D%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20%20%3D%203%20%7B%7B%5Crm%20B%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" alt="2 {{rm B}{rm M}}limits^ to = 3 {{rm B}Gamma }limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaxacabaGaeuOKdiKaeuiNd0ealeqabaGaeyOKH4kaaOGaeyypa0JaaG4mamaaxacabaGaeuOKdiKaeu4KdCealeqabaGaeyOKH4kaaaaa!4292!" /> </span></p>
<p><span style="font-size: 12pt;">Α. Να εκφραστούν τα διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%5CDelta%20%7D%5Climits%5E%20%5Cto%20%20%2C%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20E%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm A}Delta }limits^ to ,,,, {{rm A}{rm E}}limits^ to ,,,kappa alpha iota ,,, {{rm A}{rm M}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHuoaraSqabeaacqGHsgIRaaGccaGGSaGaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHroqqcqqHvoqraSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7iiaacqWF6oWAcqWFXoqycqWF5oqAcaaMc8UaaGPaVlaaykW7daWfGaqaaiabfg5abjabfY5anbWcbeqaaiabgkziUcaaaaa!58A2!" /> ως προς <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%5C%2C%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm A}{rm B},}limits^ to ,,,kappa alpha iota ,,, {{rm A}Gamma }limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqcaaMc8oaleqabaGaeyOKH4kaaOGaaGPaVlaaykW7caaMc8occaGae8NUdSMae8xSdeMae8xUdKMaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHroqqcqqHtoWraSqabeaacqGHsgIRaaaaaa!4FC2!" /></span></p>
<p><span style="font-size: 12pt;">Β. Να εκφραστούν τα διανύσματα <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%5CDelta%20%7B%5Crm%20E%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%5CDelta%20%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20" alt=" {Delta {rm E}}limits^ to ,,,kappa alpha iota ,,, {Delta {rm M}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHuoarcqqHvoqraSqabeaacqGHsgIRaaGccaaMc8UaaGPaVlaaykW7ieaajugWbiaa-P7acaWFXoGaa8xUdOGaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHuoarcqqHCoqtaSqabeaacqGHsgIRaaaaaa!4E4F!" /> ως προς <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20A%7D%7B%5Crm%20B%7D%5C%2C%7D%5Climits%5E%20%5Cto%20%20%5C%2C%5C%2C%5C%2C%5Ckappa%20%5Calpha%20%5Ciota%20%5C%2C%5C%2C%5C%2C%20%7B%7B%5Crm%20A%7D%5CGamma%20%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm A}{rm B},}limits^ to ,,,kappa alpha iota ,,, {{rm A}Gamma }limits^ to " longdesc="http://ematha.vassiliadis.edu.gr/MathType!MTEF!2!1!+-feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0df9yqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHroqqcqqHsoGqcaaMc8oaleqabaGaeyOKH4kaaOGaaGPaVlaaykW7caaMc8occaGae8NUdSMae8xSdeMae8xUdKMaaGPaVlaaykW7caaMc8+aaCbiaeaacqqHroqqcqqHtoWraSqabeaacqGHsgIRaaaaaa!4FC2!" /></span></p>
<p><span style="font-size: 12pt;">Γ. Να αποδειχτεί ότι το σημείο Ε είναι το μέσο του ΔΜ.</span></p>
12. Β΄Λυκείου/ Κατεύθυνση/ Διανύσματα
2014-10-09T18:52:39+03:00
2014-10-09T18:52:39+03:00
https://ematha.vassiliadis.edu.gr/index.php/2012-11-27-12-21-20/2012-11-27-19-25-06/dianysmata/141-12-%CE%B2%CE%84%CE%BB%CF%85%CE%BA%CE%B5%CE%AF%CE%BF%CF%85-%CE%BA%CE%B1%CF%84%CE%B5%CF%8D%CE%B8%CF%85%CE%BD%CF%83%CE%B7-%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1
Παρίσης Γιώργος
parisis@vas.edu.gr
<p><span style="font-size: 12pt;">Δίνεται τα διαφορετικά ανά δύο σημεία Α, Β και Μ και ο πραγματικός αριθμός <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Clambda%20%20%5Cin%20%5Cleft%28%20%7B0%2C1%7D%20%5Cright%29" alt="lambda in left( {0,1} right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyicI48aaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa!3C60!" />.</span></p>
<p><span style="font-size: 12pt;">Αν ισχύει η σχέση <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B2%5Clambda%20%20-%201%7D%20%5Cright%29%20%7B%7B%5Crm%20O%7D%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20%20%3D%20%5Clambda%20%20%7B%7B%5Crm%20O%7D%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20%20%2B%20%5Cleft%28%20%7B%5Clambda%20%20-%201%7D%20%5Cright%29%20%7B%7B%5Crm%20O%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20" alt="left( {2lambda - 1} right) {{rm O}{rm M}}limits^ to = lambda {{rm O}{rm A}}limits^ to + left( {lambda - 1} right) {{rm O}{rm B}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIYaGaeq4UdWMaeyOeI0IaaGymaaGaayjkaiaawMcaamaaxacabaGaeu4Nd8KaeuiNd0ealeqabaGaeyOKH4kaaOGaeyypa0Jaeq4UdW2aaCbiaeaacqqHFoWtcqqHroqqaSqabeaacqGHsgIRaaGccqGHRaWkdaqadaqaaiabeU7aSjabgkHiTiaaigdaaiaawIcacaGLPaaadaWfGaqaaiabf+5apjabfk5acbWcbeqaaiabgkziUcaaaaa!52FF!" /> να αποδειχτεί ότι:</span> </p>
<p><span style="font-size: 12pt;">Α. <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Clambda%20%20%5Cne%20%5Cfrac%7B1%7D%7B2%7D" alt="lambda ne frac{1}{2}" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyiyIK7aaSaaaeaacaaIXaaabaGaaGOmaaaaaaa!3A7C!" /> </span></p>
<p><span style="font-size: 12pt;">Β. <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20M%7D%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%2F%2F%5C%2C%5C%2C%20%7B%7B%5Crm%20M%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm M}{rm A}}limits^ to ,//,, {{rm M}{rm B}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHCoqtcqqHroqqaSqabeaacqGHsgIRaaGccaaMc8Uaai4laiaac+cacaaMc8UaaGPaVpaaxacabaGaeuiNd0KaeuOKdiealeqabaGaeyOKH4kaaaaa!45A7!" /> </span></p>
<p><span style="font-size: 12pt;">Γ. Το σημείο Μ είναι εξωτερικό του τμήματος ΑΒ.</span></p>
<p><span style="font-size: 12pt;">Δίνεται τα διαφορετικά ανά δύο σημεία Α, Β και Μ και ο πραγματικός αριθμός <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Clambda%20%20%5Cin%20%5Cleft%28%20%7B0%2C1%7D%20%5Cright%29" alt="lambda in left( {0,1} right)" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyicI48aaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa!3C60!" />.</span></p>
<p><span style="font-size: 12pt;">Αν ισχύει η σχέση <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%28%20%7B2%5Clambda%20%20-%201%7D%20%5Cright%29%20%7B%7B%5Crm%20O%7D%7B%5Crm%20M%7D%7D%5Climits%5E%20%5Cto%20%20%20%3D%20%5Clambda%20%20%7B%7B%5Crm%20O%7D%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20%20%2B%20%5Cleft%28%20%7B%5Clambda%20%20-%201%7D%20%5Cright%29%20%7B%7B%5Crm%20O%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20" alt="left( {2lambda - 1} right) {{rm O}{rm M}}limits^ to = lambda {{rm O}{rm A}}limits^ to + left( {lambda - 1} right) {{rm O}{rm B}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacaaIYaGaeq4UdWMaeyOeI0IaaGymaaGaayjkaiaawMcaamaaxacabaGaeu4Nd8KaeuiNd0ealeqabaGaeyOKH4kaaOGaeyypa0Jaeq4UdW2aaCbiaeaacqqHFoWtcqqHroqqaSqabeaacqGHsgIRaaGccqGHRaWkdaqadaqaaiabeU7aSjabgkHiTiaaigdaaiaawIcacaGLPaaadaWfGaqaaiabf+5apjabfk5acbWcbeqaaiabgkziUcaaaaa!52FF!" /> να αποδειχτεί ότι:</span> </p>
<p><span style="font-size: 12pt;">Α. <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Clambda%20%20%5Cne%20%5Cfrac%7B1%7D%7B2%7D" alt="lambda ne frac{1}{2}" longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyiyIK7aaSaaaeaacaaIXaaabaGaaGOmaaaaaaa!3A7C!" /> </span></p>
<p><span style="font-size: 12pt;">Β. <img class="ee_img tr_noresize" src="http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%20%7B%7B%5Crm%20M%7D%7B%5Crm%20A%7D%7D%5Climits%5E%20%5Cto%20%20%5C%2C%2F%2F%5C%2C%5C%2C%20%7B%7B%5Crm%20M%7D%7B%5Crm%20B%7D%7D%5Climits%5E%20%5Cto%20%20" alt=" {{rm M}{rm A}}limits^ to ,//,, {{rm M}{rm B}}limits^ to " longdesc="MathType!MTEF!2!1!+-feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipD0dg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacqqHCoqtcqqHroqqaSqabeaacqGHsgIRaaGccaaMc8Uaai4laiaac+cacaaMc8UaaGPaVpaaxacabaGaeuiNd0KaeuOKdiealeqabaGaeyOKH4kaaaaa!45A7!" /> </span></p>
<p><span style="font-size: 12pt;">Γ. Το σημείο Μ είναι εξωτερικό του τμήματος ΑΒ.</span></p>