Question

A, B have position vectors $\overrightarrow{a},\overrightarrow{b}$ relative to the origin O and X, Y divide $\overrightarrow{AB}$ internally and externally respectively in the ratio 2 : 1. Then,$\overrightarrow{XY}$ is equal to

• A. $\frac{3}{2}(\overrightarrow{b}-\overrightarrow{a})$
• B. $\frac{4}{3}(\overrightarrow{a}-\overrightarrow{b})$
• C. $\frac{5}{6}(\overrightarrow{b}-\overrightarrow{a})$
• D. $\frac{4}{3}(\overrightarrow{b}-\overrightarrow{a})$

Answer 1

The correct option is D $\frac{4}{3}(\overrightarrow{b}-\overrightarrow{a})$

Since X and Y divide $\overrightarrow{AB}$ internally and externally in the ratio 2 : 1, therefore the position vectors of X and Y are given by $\frac{2\overrightarrow{b}+\overrightarrow{a}}{3}$ and $2\overrightarrow{b}-\overrightarrow{a}$ respectively.
Thus, $\overrightarrow{XY}=(2\overrightarrow{b}-\overrightarrow{a})-\frac{1}{3}(2\overrightarrow{b}+\overrightarrow{a})=\frac{4}{3}(\overrightarrow{b}-\overrightarrow{a})$