Question

Equation of the perpendicular bisector of the line joining the points whose position vectors are $\vec{a}$ and $\vec{b}$ respectively is

• A. $(2\vec{r}-\vec{a}-\vec{b})\cdot (\vec{a}-\vec{b})=0$
• B. $(2\vec{r}-\vec{a}-\vec{b})\cdot (\vec{a}+\vec{b})=0$
• C. $(2\vec{r}+\vec{a}+\vec{b})\cdot (\vec{a}-\vec{b})=0$
• D. $(2\vec{r}-\vec{a}+\vec{b})\cdot (\vec{a}+\vec{b})=0$

Answer 1

Answer: (A)

The mid-point of line joining points whose position vectors are $a$ and $b$ is $M\left(\frac{a+b}{2}\right)$ and the direction ratio vector of line joining of given points is $(a-b)$

Let a variable point $p(r)$ on the perpendicular bisector of $AB$, so $MP\perp BA$
$\Rightarrow \left(r-\frac{a+b}{2}\right)\cdot (a-b)=0$
$\Rightarrow (2r-a-b)\cdot (a-b)=0$