Question

Find a position vector of a point $R$ which devides 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 $2\ratio 1$
(i)internally
(ii)externally

Answer 1

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:
$\vec{OP}=\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+\hat{j}+\hat{k}$
(i)The position vector of point $R$ which divides the line joining two points $P$ and $Q$ internally in the ratio $2\ratio 1$ is given by,
$\vec{OR}=\frac{2(-\hat{i}+\hat{j}+\hat{k})+1(\hat{i}+2\hat{j}-\hat{k})}{2+1}$
$=\frac{(-2\hat{i}+2\hat{j}+2\hat{k})+(\hat{i}+2\hat{j}-\hat{k})}{3}$
$=\frac{-\hat{i}+4\hat{j}+\hat{k}}{3}$
$=\frac{-1}{3}\hat{i}+\frac{4}{3}\hat{j}+\frac{1}{3}\hat{k}$
(ii)The position vector of point $R$ which devides the line joining $P$ and $Q$ externally in the ratio $2\ratio 1$ is given by,
$\vec{OR}=\frac{2(-\hat{i}+\hat{j}+\hat{k})-1(\hat{i}+2\hat{j}-\hat{k})}{2-1}$
$=(-2\hat{i}+2\hat{j}+2\hat{k})-(\hat{i}+2\hat{j}-\hat{k})$
$=-3\hat{i}+3\hat{k}$.