Find the position vector of a point

The position vector of point R dividing the line segment joining two points

P and Q in the ratio m: n is given by:

i. Internally:

$\frac{m \vec{b}+n \vec{a}}{m+n}$

ii. Externally:

$\frac{m \vec{b}-n \vec{a}}{m-n}$

Position vectors of P and Q are given as:

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

(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,

$\overrightarrow{\mathrm{OR}}=\frac{2(-\hat{i}+\hat{j}+\hat{k})+1(\hat{i}+2 \hat{j}-\hat{k})}{2+1}=\frac{(-2 \hat{i}+2 \hat{j}+2 \hat{k})+(\hat{i}+2 \hat{j}-\hat{k})}{3}$

$=\frac{-\hat{i}+4 \hat{j}+\hat{k}}{3}=-\frac{1}{3} \hat{i}+\frac{4}{3} \hat{j}+\frac{1}{3} \hat{k}$

(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by,

$\overrightarrow{\mathrm{OR}}=\frac{2(-\hat{i}+\hat{j}+\hat{k})-1(\hat{i}+2 \hat{j}-\hat{k})}{2-1}=(-2 \hat{i}+2 \hat{j}+2 \hat{k})-(\hat{i}+2 \hat{j}-\hat{k})$

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