Question

Prove $:|\overline{a}+\overline{b}|=|\overline{a}-\overline{b}|\iff \overline{a}\perp \overline{b}$ . Interpret this geometrically.

Answer 1

Consider given the vectors equation,

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

Taking square both sides,

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

$\because {\left|\overrightarrow{x}\right|}^2={\overrightarrow{x}}^2$

there fore

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

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

$2.\overrightarrow{a}.\overrightarrow{b}=-2.\overrightarrow{a}.\overrightarrow{b}$

$4\overrightarrow{a}.\overrightarrow{b}=0$

$\overrightarrow{a}.\overrightarrow{b}=0$

$\therefore \overrightarrow{a}\perp \overrightarrow{b}$

Hence, this is the answer.