Question

$\overline{a}$ and $\overline{b}$ are the position vectors of A and B respectively. Points P and Q divide AB, internally and externally in the same ratio 2:1 , then PQ is equal to

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

Answer 1

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

Position vector of P will be $\overrightarrow{OP}=\left(\frac{2\overline{b}+\overline{a}}{3}\right)$ (P divides AB internally in the ratio $2:1$)

Position vector of Q will be $\overrightarrow{OQ}=\left(\frac{2\overline{b}-\overline{a}}{2-1}\right)=\left(\frac{2\overline{b}-\overline{a}}{1}\right)$

(Q divides AB externally in the ratio $2:1$)

$\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=(2\overrightarrow{b}-\overrightarrow{a})-\left(\frac{2\overline{b}+\overline{a}}{3}\right)=\frac{4\overrightarrow{b}}{3}-\frac{4\overrightarrow{a}}{3}$

$=\frac{4}{3}(\overrightarrow{b}-\overrightarrow{a})$.