Question

If $\overrightarrow{a}$, $\overrightarrow{b}$ are two vectors such that $\left|\overrightarrow{a}+\overrightarrow{b}\right|=\left|\overrightarrow{b}\right|$, then prove that $\overrightarrow{a}+2\overrightarrow{b}$is perpendicular to $\overrightarrow{a}$.

Answer 1

$$\begin{gathered} \text{Given that} \\ \left|\overrightarrow{a}+\overrightarrow{b}\right|=\left|\overrightarrow{b}\right| \\ \text{Squaring both sides, we get} \\ {\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2={\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}={\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}=0...\left(1\right) \\ \text{Now,} \\ \left(\overrightarrow{a}+2\overrightarrow{b}\right).\overrightarrow{a} \\ =\overrightarrow{a}.\overrightarrow{a}+2\overrightarrow{b}.\overrightarrow{a} \\ ={\left|\overrightarrow{a}\right|}^2+2\overrightarrow{a}.\overrightarrow{b} \\ =0\left[\mathrm{Using}\left(1\right)\right] \\ \text{So,}\overrightarrow{a}+2\overrightarrow{b}\text{is perpendicular to}\overrightarrow{a}\text{.} \end{gathered}$$