Question

If a vector $\overset{ \rightarrow }{r}$ of magnitude $3\sqrt{6}$ is directed along the bisector of the angle between the vectors $\overset{ \rightarrow }{a}=7\hat{i}-4\hat{j}-4\hat{k}$ and $\overset{ \rightarrow }{b}=-2\hat{i}-\hat{j}+2\hat{k}$ , then $\overset{ \rightarrow }{r}$ can be

• 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. $\hat{i}-7\hat{j}-2\hat{k}$

Answer 1

Answer: (A)

Given, $\overset{ \rightarrow }{a}=7\hat{i}-4\hat{j}-4\hat{k},\overset{ \rightarrow }{b}=-2\hat{i}-\hat{j}+2\hat{k},\left|\overset{ \rightarrow }{r}\right|=3\sqrt{6}$
and $\overset{ \rightarrow }{r}$ is the bisector of the angle between $\overset{ \rightarrow }{a}\&\overset{ \rightarrow }{b}.$
$\therefore \overset{ \rightarrow }{r}=\lambda \left(\frac{\overset{ \rightarrow }{a}}{\left|\overset{ \rightarrow }{a}\right|} + \frac{\overset{ \rightarrow }{b}}{\left|\overset{ \rightarrow }{b}\right|}\right)$
$\Rightarrow \overset{ \rightarrow }{r}=\lambda \left(\frac{\left(7 \hat{i} - 4 \hat{j} - 4 \hat{k}\right)}{\left|7 \hat{i} - 4 \hat{j} - 4 \hat{k}\right|} + \frac{\left(- 2 \hat{i} - \hat{j} + 2 \hat{k}\right)}{\left|- 2 \hat{i} - \hat{j} + 2 \hat{k}\right|}\right)$
$\Rightarrow \overset{ \rightarrow }{r}=\lambda \left(\frac{\left(7 \hat{i} - 4 \hat{j} - 4 \hat{k}\right)}{\sqrt{7^{2} + \left(- 4\right)^{2} + \left(- 4\right)^{2}}} + \frac{\left(- 2 \hat{i} - \hat{j} + 2 \hat{k}\right)}{\sqrt{\left(- 2\right)^{2} + \left(- 1\right)^{2} + 2^{2}}}\right)$
$\Rightarrow \overset{ \rightarrow }{r}=\lambda \left(\frac{\left(7 \hat{i} - 4 \hat{j} - 4 \hat{k}\right)}{9} + \frac{\left(- 2 \hat{i} - \hat{j} + 2 \hat{k}\right)}{3}\right)$
$\Rightarrow \overset{ \rightarrow }{r}=\frac{\lambda }{9}\left(\hat{i} - 7 \hat{j} + 2 \hat{k}\right)...\left(i\right)$
$\therefore \left|\overset{ \rightarrow }{r}\right|=\sqrt{\left(\frac{\lambda }{9}\right)^{2} \left(1^{2} + \left(- 7\right)^{2} + \left(2\right)^{2}\right)}=\frac{\lambda }{9}\sqrt{54}...\left(i i\right)$
Also, given $\left|\overset{ \rightarrow }{r}\right|=3\sqrt{6}...\left(i i i\right)$
From $\left(i i\right)\&\left(i i i\right)$ we get,
$3\sqrt{6}=\frac{\lambda }{9}\sqrt{54}$
$\Rightarrow \frac{54 \lambda ^{2}}{81}=54$
$\Rightarrow \lambda =\pm9$
Putting in the equation $\left(i\right)$ , we get
$\overset{ \rightarrow }{r}=\frac{\pm 9}{9}\left(\hat{i} - 7 \hat{j} + 2 \hat{k}\right)$
$\Rightarrow \overset{ \rightarrow }{r}=\pm\left(\hat{i} - 7 \hat{j} + 2 \hat{k}\right)$
According to the options, $\overset{ \rightarrow }{r}=\hat{i}-7\hat{j}+2\hat{k}$ .