PYQ NEET- Motion In A Plane L-2

Question: If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is#

A) $90^{\circ}$

B) $45^{\circ}$

C) $180^{\circ}$

D) $0^{\circ}$

Answer: $90^{\circ}$#

Sol:#

Suppose two vectors are $\mathbf{P}$ and $\mathbf{0}$. It is given that $$ |\mathbf{P}+\mathbf{0}|=|\mathbf{P}-\mathbf{0}| $$

Let angle between $\mathbf{P}$ and $\mathbf{0}$ be $\boldsymbol{\phi}$. $$ \begin{alignedat} & \therefore \ & P^2+Q^2+2 P Q \cos \phi=P^2+Q^2-2 P Q \cos \phi \ & \Rightarrow \quad 4 P Q \cos \phi=0 \ & \Rightarrow \quad \quad \cos \phi=0 \quad[\because P, Q \neq 0] \ & \Rightarrow \quad \phi=\frac{\pi}{2}\ \text{rad}=90^{\circ} \end{aligned} $$