Question

If $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$, then which one of the following is correct?

• A. $|\overrightarrow{a}|=|\overrightarrow{b}|$
• B. $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$
• C. $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$
• D. $\overrightarrow{a}$ is a unit vector

Answer 1

Answer: (A), (C)

$|\overrightarrow{a}|=|\overrightarrow{b}|$
$\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$

Here given $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$

$\Rightarrow ${|\overrightarrow{a}+\overrightarrow{b}|}^2={|\overrightarrow{a}-\overrightarrow{b}|}^2$

$\to x^2+y^2+2xy\cos \theta =x^2+y^2-2xycos\theta$

$\cos \theta =-\cos \theta$

$\cos \theta =0$

$\theta ={90}^o$

Hence here $\overrightarrow{a}$ is perpedicular to $\overrightarrow{b}$ and $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|$.