Question

The vector $\overrightarrow{C}$, directed along the internal bisector of the angle between the vectors $\overrightarrow{a}=7\hat{i}-4\hat{j}-4\hat{k}$ and $\overrightarrow{b}=-2\hat{i}-\hat{j}+2\hat{k}$ with $|\overrightarrow{c}|=5\sqrt{6}$, is

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

Answer 1

The correct option is A $\pm \left(\frac{5}{3}\right(\hat{i}-7\hat{j}+2\hat{k})$

The required vector $\overrightarrow{C}$ is given by
$\overrightarrow{C}=\lambda (\hat{a}+\hat{b})=\lambda (\frac{\overrightarrow{a}}{|\overrightarrow{a}|}+\frac{\overrightarrow{b}}{|\overrightarrow{b}|})=\lambda \left\{\frac{1}{9}\right(7\hat{i}-4\hat{j}-4\hat{k})+\frac{1}{3}(-2\hat{i}-\hat{j}+2\hat{k}\left)\right\}$
$\Rightarrow \overrightarrow{c}=\frac{\lambda }{9}(\hat{i}-7\hat{j}+2\hat{k})\Rightarrow |\overrightarrow{c}|=\pm \frac{\lambda }{9}\sqrt{1+49+4}=\pm \frac{\lambda }{9}\sqrt{54}$
But $|\overrightarrow{c}|=5\sqrt{6}$ (given)
$\therefore \pm \frac{\lambda }{9}\sqrt{54}=5\sqrt{6}\Rightarrow \lambda =\pm 15$
Hence, $\overrightarrow{c}=\pm \frac{15}{9}(\hat{i}-7\hat{j}+2\hat{k})=\pm \frac{5}{3}(\hat{i}-7\hat{j}+2\hat{k})$