Find 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 $\overrightarrow{b}$ = $-\hat{i} + \hat{j} + \hat{k} $ respectively, in the ratio 2:1 (i) internally (ii) externally.

The correct answer is (i) Internally $\dfrac{-\hat{i} + 4\hat{j} + \hat{k}}{3}$
(ii) Externally $-3\hat{i} + 3\hat{k}$

Let $\overrightarrow{OP} = \hat{i} + 2\hat{j} - \hat{k}$ and $\overrightarrow{OQ}$ = $-\hat{i} + \hat{j} + \hat{k} $

(i) Internally $\overrightarrow{OR} = \dfrac{\overrightarrow{OP} + 2\overrightarrow{OQ}}{1 + 2}$ = $\dfrac{-\hat{i} + 4\hat{j} + \hat{k}}{3}$

(i) Externally $\overrightarrow{OR} = \dfrac{-\overrightarrow{OP} + 2\overrightarrow{OQ}}{2 - 1}$ = $-3\hat{i} + 3\hat{k}$