Question

If $|\overrightarrow{a}|$ and $|\overrightarrow{b}|$ are equal and both are perpendicular then angle between $\overrightarrow{a}$ and $|\overrightarrow{a}+\overrightarrow{b}|$ is

• A. ${45}^o$
• B. ${42}^o$
• C. ${60}^o$
• D. ${70}^o$

Answer 1

The correct option is A ${45}^o$

Given $|\overrightarrow{a}|=|\overrightarrow{b}|$, $\overrightarrow{a}.\overrightarrow{b}=0$,

$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}{|}^2=(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}+\overrightarrow{b})=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2=2|\overrightarrow{a}{|}^2$

$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}|=\sqrt{2}|\overrightarrow{a}|$

Let angle between $\overrightarrow{a}$ and $(\overrightarrow{a}+\overrightarrow{b})$ is $\theta$.

$\Rightarrow \overrightarrow{a}.(\overrightarrow{a}+\overrightarrow{b})=|\overrightarrow{a}||\overrightarrow{a}+\overrightarrow{b}|cos\left(\theta \right)$

$\Rightarrow |\overrightarrow{a}{|}^2+\overrightarrow{a}.\overrightarrow{b}=\sqrt{2}|\overrightarrow{a}{|}^{2}cos\left(\theta \right)$

$\Rightarrow cos\left(\theta \right)=\frac{1}{\sqrt{2}}$ $\Rightarrow \theta ={45}^o$