Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $^i + 2^j -^k$ and $-^i +^j +^k$ respectively, in the ratio $2 : 1$ ,
(i) internally
(ii) externally

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Solution

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:
$\frac{m \to Q + n \to P}{m + n}$
ii. Externally:
$\frac{m \to Q - n \to P}{m - n}$
Position vectors of $P$ and $Q$ are given as:
$\to O P =^i + 2^j -^k$ and $\to O Q = -^i +^j +^k$
(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,
$\to O R = \frac{2 (-^i +^j +^k) + 1 (^i + 2^j -^k)}{2 + 1} = \frac{- 2^i + 2^j +^k + (^i + 2^j -^k)}{3}$
$= \frac{-^i + 4^j +^k}{3} = - \frac{1}{3}^i + \frac{4}{3}^j + \frac{1}{3}^k$
(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,
$\to O R = \frac{2 (-^i +^j +^k) - 1 (^i + 2^j -^k)}{2 - 1} = (- 2^i + 2^j + 2^k) - (^i + 2^j -^k)$
$= - 3^i + 3^k$

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