Question

If $|\overline{a}+\overline{b}|=|\overline{a}-\overline{b}|,\overline{a},\overline{b}\ne 0,$ then

• A. $\overline{a}$ is $\left|\right|$ to $\overline{b}$
• B. $\overline{a}$ is $\perp$ to $\overline{b}$
• C. $\left|a\right|=\left|b\right|$
• D. None of these

Answer 1

The correct option is B $\overline{a}$ is $\perp$ to $\overline{b}$

If $|\overline{a}+\overline{b}|=|\overline{a}-\overline{b}|,\overline{a},\overline{b}\ne 0,$
Squaring on both sides
${|\overline{a}+\overline{b}|}^2={|\overline{a}-\overline{b}|}^2$
$\Rightarrow |a{|}^2+|b{|}^2+2\overline{a}\cdot \overline{b}=a^2+b^2-2\overline{a}\cdot \overline{b}$
$\Rightarrow \overline{a}\cdot \overline{b}=0$
$\therefore \overline{a}$ and $\overline{b}$ are perpendicular.
Hence, option B.