Question

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors \[\vec{OP} = 2 \vec{a} + \vec{b}\] and \[\vec{OQ} = \vec{a} - 2 \vec{b}\], respectively in the ratio 1 : 2 internally and externally. 

Answer 1

It is given that P and Q are two points with position vectors\[\vec{OP} = 2 \vec{a} + \vec{b}\] and \[\vec{OP} = 2 \vec{a} + \vec{b}\], respectively.
When R divides PQ internally in the ratio 1 : 2, then
Position vector of R = \[\frac{1 \times \left( \vec{a} - 2 \vec{b} \right) + 2 \times \left( 2 \vec{a} + \vec{b} \right)}{1 + 2} = \frac{5 \vec{a}}{3}\]
When R divides PQ externally in the ratio 1 : 2, then Position vector of R =\[\frac{1 \times \left( \vec{a} - 2 \vec{b} \right) - 2 \times \left( 2 \vec{a} + \vec{b} \right)}{1 - 2} = \frac{- 3 \vec{a} - 4 \vec{b}}{- 1} = 3 \vec{a} + 4 \vec{b}\]