સદીશ , c એસદીશો a= ,7i ,- , ,4j , ,-4k અને b =- ,2i ,- , ,j , ,+2… - Vidyadip

MCQ

સદીશ $\,\vec c$ એસદીશો $\vec a=\,7\hat{i}\,-\,\,4\hat{j}\,\,-4\hat{k}$ અને $\vec b =-\,2\hat{i}\,-\,\,\hat{j}\,\,+2\hat{k}$ વ્ચ્ચેના ખૂણાના અંત : સમવિભાજકની દિશામાં $|\vec c|\,\,=\,\,5\sqrt{6,}$ સાથે હોય તો સદીશ $\vec c$ મેળવો.

A
$\frac{5}{3}\,\,\left( {\hat i\, - \,\,7\hat j\,\, + 2\hat k\,} \right)$
B
$\frac{5}{3}\,\,\left( {5\hat i\, + \,\,5\hat j\,\, + 2\hat k\,} \right)$
C
$\frac{5}{3}\,\,\left( {\hat i\, + \,\,7\hat j\,\, + 2\hat k\,} \right)$
D
$\frac{5}{3}\,\,\left( { - 5\hat i\, + \,\,5\hat j\,\, + 2\hat k\,} \right)$

✓

Answer

માંગેલ સદીશ $\text{c}$ એ $\lambda \,\,\,\left( \frac{\text{a}}{\text{ }\!\!|\!\!\text{ a }\!\!|\!\!\text{ }}\,\,+\ \,\frac{b}{|b|} \right)$ દ્વારા દર્શાવાય.

હવે $\frac{a}{|a|}\,\,=\,\,\frac{1}{9}\,\,\left( 7i\,\,-\,\,4j\,\,-\,\,4k \right)$ અને $\frac{b}{|b|}\,\,=\,\,\frac{1}{3}\,\,\left( -2\hat{i}-\,\,\hat{j}\,\,+\,\,2\hat{k} \right)$

$\Rightarrow \,\,c\,\,=\,\,\lambda \,\,\left( \frac{1}{9}\hat{i}-\,\,\frac{7}{9}\hat{j}\,\,+\,\,\frac{2}{3}\hat{k} \right)\,\,$

$|c{{|}^{2}}\,\,=\,\,{{\lambda }^{2}}\,\,\frac{54}{81}$

${{\lambda }^{2}}\,\,=\,\,225$

$\lambda \,=\,\,\pm \,\,15$

તેથી $c\,\,=\,\,\pm \frac{5}{3}\,\,\left( \hat{i}-\,7\hat{j}\,\,+\,\,2\hat{k} \right)$

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