Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors inclined at an angle $\theta$ such that $\overrightarrow{a}+\overrightarrow{b}$ is a unit vector, then $\theta$ is equal to

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is D $\frac{2\pi }{3}$

It is given that, $|\overrightarrow{a}|=|\overrightarrow{b}|=|\overrightarrow{a}+\overrightarrow{b}|=1$
We have, $|\overrightarrow{a}+\overrightarrow{b}{|}^2=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2+2\overrightarrow{a}.\overrightarrow{b}$
$\Rightarrow 1=1+1+2|\overrightarrow{a}||\overrightarrow{b}|cos\theta \Rightarrow cos\theta =-\frac{1}{2}\Rightarrow \theta =\frac{2\pi }{3}$