Question

The position vector of the points $A$ and $B$ are respectively $\overline{a}$ and $\overline{b}$ divides $AB$ in the ratio $3:1$ and $Q$ iis the midpoint of $AP$. The position vector of $Q$ is

• A. $\frac{3\overline{a}+5\overline{b}}{8}$
• B. $\frac{3\overline{a}+\overline{b}}{4}$
• C. $\frac{\overline{a}+3\overline{b}}{4}$
• D. $\frac{5\overline{a}+3\overline{b}}{8}$

Answer 1

The correct option is D $\frac{5\overline{a}+3\overline{b}}{8}$

Given

position vector of A

$\overrightarrow{OA}=\overrightarrow{a}$

position vector of B

$\overrightarrow{OB}=\overrightarrow{b}$

Point P divides AB in ration $3:1$

position vector of Point P

$\overrightarrow{OP}=\frac{3\overline{OB}+\overline{OA}}{4}$

$\overrightarrow{OP}=\frac{3\overline{b}+\overline{a}}{4}$

Here Q is the mid point of AP

SO $\overrightarrow{OQ}=\frac{\overrightarrow{OA}+\overrightarrow{OP}}{2}$

$Q=\frac{\frac{3\overline{b}+\overline{a}}{4}+\overline{a}}{2}=\frac{5\overline{a}+3\overline{b}}{8}$