Question

Let $\overrightarrow{OA}=\hat{i}+3\hat{j}-2\hat{k}$ and $\overrightarrow{OB}=3\hat{i}+\hat{j}-2\hat{k}.$
The vector $\overrightarrow{OC}$ bisecting the angle $AOB$ and $C$ being the point on the line $AB$ is

• A. $4(\hat{i}+\hat{j}-\hat{k})$
• B. $2(\hat{i}+\hat{j}-\hat{k})$
• C. $(\hat{i}+\hat{j}-\hat{k})$
• D. None of these

Answer 1

The correct option is B $2(\hat{i}+\hat{j}-\hat{k})$

$\left|\overrightarrow{OA}\right|=\sqrt{14},\left|\overrightarrow{OB}\right|=\sqrt{14}$
So, the bisector $OC$ of $\angle AOB$ must intersect $AB$ at its mid point $C$
$\therefore \overrightarrow{OC}=\frac{1}{2}(\overrightarrow{OA}+\overrightarrow{OB})$
$=\frac{1}{2}(\hat{i}+3\hat{j}-2\hat{k}+3\hat{j}+\hat{j}-2\hat{k})$
$=\frac{1}{2}(4\hat{i}+4\hat{j}-4\hat{k})$
$=2(\hat{i}+\hat{j}-\hat{k})$

Hence, option B.