Question

$P$ and $Q$ are two points with position vectors $3\overrightarrow{a}-2\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$ respectively. Write the position vector of a point R which divides the line segment $PQ$ in the ratio $2:1$ externally.

Answer 1

Position vector of $P=$ $3\overrightarrow{a}-2\overrightarrow{b}$

Position vector of $Q=$ $\overrightarrow{a}+\overrightarrow{b}$

Point R divides segment PQ in ratio 2:1 externally.

Position vector of $R=$ $\frac{\left(PositionofP\right)1-\left(PositionofQ\right)2}{1-2}$

Position vector of $R=$ $\frac{(3\overrightarrow{a}-2\overrightarrow{b})1-(\overrightarrow{a}+\overrightarrow{b})2}{-1}=\frac{\overrightarrow{a}-4\overrightarrow{b}}{-1}$

Therefore, position vector of $R=$ $4\overrightarrow{b}-\overrightarrow{a}.$