Question

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