The position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively in the ratio 2 : 1 externally is :

$-3\hat i+3\hat k$

The correct answer is option (2) → $-3\hat i+3\hat k$

For external division, the formula is:

$\vec{R} = \frac{m\vec{Q} - n\vec{P}}{m - n}$

Given:

$\vec{P} = \hat{i} + 2\hat{j} - \hat{k}$

$\vec{Q} = -\hat{i} + \hat{j} + \hat{k}$

Ratio $2 : 1$ (externally)

Substitution

$ \vec{R} = \frac{2(-\hat{i} + \hat{j} + \hat{k}) - 1(\hat{i} + 2\hat{j} - \hat{k})}{2 - 1}$

$= \frac{-2\hat{i} + 2\hat{j} + 2\hat{k} - \hat{i} - 2\hat{j} + \hat{k}}{1}$

$= -3\hat{i} + 0\hat{j} + 3\hat{k}$