Find the position vector of a point $\mathrm{R}$ which divides the line joining two points $\mathrm{P}$ and $\mathrm{Q}$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively, in the ration $2: 1$
(i) internally
(ii) externally
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,
$\overrightarrow{\mathrm{OR}}=\frac{2(-\hat{i}+\hat{j}+\hat{k})-1(\hat{i}+2 \hat{j}-\hat{k})}{2-1}=(-2 \hat{i}+2 \hat{j}+2 \hat{k})-(\hat{i}+2 \hat{j}-\hat{k})$
$=-3 \hat{i}+3 \hat{k}$