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 $A B$, so $M P \perp B A$
$\Rightarrow \left(r-\frac{a+ b}{2}\right) \cdot(a-b)=0$
$\Rightarrow (2 r-a-b) \cdot(a-b)=0$