Question

If $\overline{a}+\overline{b}$ is perpendicular to $\overline{b}$ and $\overline{a}+2\overline{b}$ perpendicular to $\overline{a}$ then

• A. $|\overline{a}|=|\overline{b}|$
• B. $|\overline{a}|=\sqrt{2}|\overline{b}|$
• C. $|\overline{b}|=\sqrt{2}|\overline{b}|$
• D. $|\overline{a}|=|\overline{b}|\sqrt{3}$

Answer 1

The correct option is B $|\overline{a}|=\sqrt{2}|\overline{b}|$

$\begin{array}{c}\left(\overrightarrow{a}+\overrightarrow{b}\right).\overrightarrow{b}=0and\left(\overrightarrow{a}+2\overrightarrow{b}\right).\overrightarrow{a}=0\\ \Rightarrow \overrightarrow{a}.\overrightarrow{b}+{\left|\overrightarrow{b}\right|}^2=0\Rightarrow {\left|\overrightarrow{a}\right|}^2+2\overrightarrow{b}.\overrightarrow{a}=0\\ \overrightarrow{a}.\overrightarrow{b}=-{\left|\overrightarrow{b}\right|}^2\Rightarrow \left|\overrightarrow{a}\right|=\sqrt{2}\left|\overrightarrow{b}\right|.\\ Hence,theoptionBiscorrectanswer.\end{array}$