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The position of vectors of the points P and Q with respect to the origin O are $\vec{a} = \hat{i} + 3 \hat{j} - 2 \hat{k}$ and $\vec{b} = 3 \hat{i} - \hat{j} - 2 \hat{k}$ respectively. If M is a point on PQ ,such that OM is the bisector of $∠$ POQ, then $\vec{OM}$ is

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$2 (\hat{i} - \hat{j} + \hat{k})$

$2 \hat{i} + \hat{j} - 2 \hat{k}$

$2 (- \hat{i} + \hat{j} - \hat{k})$

$2 (\hat{i} + \hat{j} + \hat{k})$

answer is B.

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Detailed Solution

Since $| \vec{OP} | = | \vec{OQ} | = \sqrt{14}, ΔOPQ$ is isosceles.
Hence, the internal bisector OM is perpendicular to PQ and M is the midpoint of P and Q.

Therefore,
$\vec{OM} = \frac{1}{2} (\vec{OP} + \vec{OQ}) = 2 \hat{i} + \hat{j} - 2 \hat{k}$

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