Question

If $\left|\overrightarrow{a}+\overrightarrow{b}\right|=\left|\overrightarrow{a}-\overrightarrow{b}\right|,$ then prove that $\overrightarrow{a}$ and $\overrightarrow{b}$ are mutually perpendicular vectors.

Answer 1

$|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\to$ Let $\theta$ be angle between them

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

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

$-2|\overrightarrow{a}||\overrightarrow{b}|\cos \theta =2|\overrightarrow{a}||\overrightarrow{b}|\cos \theta$.

$-\cos \theta =0.\cos \theta$

$\theta =\frac{\pi }{2}$

Vectors are mutually perpendicular.