The position vector of a point that divides the line segment joining $\mathrm{P} \equiv(1,2,-1)$ and…
The position vector of a point that divides the line segment joining $\mathrm{P} \equiv(1,2,-1)$ and $\mathrm{Q} \equiv(-1,1,1)$ externally in the ratio $1: 2$, is
$3 \hat{i}-3 \hat{k}$
$3 \hat{i}+3 \hat{j}-3 \hat{k}$
$-3 \hat{i}+3 \hat{k}$
$3 \hat{i}+\hat{j}+3 \hat{k}$
Solution
Using section formula for external division
$\begin{aligned} & \left(\frac{1 \times(-1)-2 \times 1}{1-2}, \frac{1 \times 1-2 \times 2}{1-2}, \frac{1 \times 1-2 \times(-1)}{1-2}\right) \equiv(3,3,-3) \\ & \equiv 3 \hat{i}+3 \hat{j}-3 \hat{k}\end{aligned}$