question_answer

The vector \[\overset{\to }{\mathop{c}}\,\] directed along the bisectors of the angle between the vectors \[\overset{\to }{\mathop{a}}\,=7\hat{i}-4\hat{j}-4\hat{k},\] \[\overset{\to }{\mathop{b}}\,=-2\hat{i}-\hat{j}+2\hat{k},\] and \[|\overset{\to }{\mathop{c}}\,|=3\sqrt{6}\] is given by

A) \[\hat{i}-7\hat{j}+2\hat{k}\]

B) \[\hat{i}+7\hat{j}-2\hat{k}\]

C) \[\hat{i}+7\hat{j}+2\hat{k}\]

D) \[\hat{i}+7\hat{j}+3\hat{k}\]

Correct Answer: A

Solution :

[a]

\[OQ=PQ=\lambda \] (say);

\[\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{QP};\overrightarrow{c}=\lambda \hat{a}+\lambda \hat{b}\]

Let \[\hat{a}\] and \[\hat{b}\] be unit

Vectors along \[\overset{\to }{\mathop{a}}\,\] and \[\overset{\to }{\mathop{b}}\,\]

Respectively,

Then \[\hat{a}=\frac{1}{9}(7\hat{i}-4\hat{j}-4\hat{k})\] and

\[\hat{b}=\frac{1}{3}(-2\hat{i}-\hat{j}+2\hat{k})\]

The required vector

\[\overset{\to }{\mathop{c}}\,=\lambda (\hat{a}+\hat{b}),\] where \[\lambda \] is a scalar

\[\lambda \left( \frac{1}{9}\hat{i}-\frac{7}{9}\hat{j}+\frac{2}{9}\hat{k} \right)\]

\[|\overset{\to }{\mathop{c}}\,{{|}^{2}}{{\lambda }^{2}}\left( \frac{1}{81}+\frac{49}{81}+\frac{4}{81} \right)=\frac{54}{81}{{\lambda }^{2}}\]

\[\Rightarrow {{(3\sqrt{6})}^{2}}=\frac{54}{81}{{\lambda }^{2}}\]

\[\Rightarrow {{\lambda }^{2}}=81\Rightarrow \lambda =\pm 9.\] Hence, \[\overset{\to }{\mathop{c}}\,=(\hat{i}-7\hat{j}+2\hat{k})\]