Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta $ is the angle between them. If $\vec{a} +\vec{b}$ is a unit vector, then $\theta $ is :

$\frac{2\pi }{3}$ $\frac{\pi }{2}$ $\frac{\pi }{3}$ $\frac{\pi }{4}$

$\frac{2\pi }{3}$

$|\vec a+\vec b|=1$ so $|\vec a|+|\vec b|+2\vec a.\vec b=1^2$ (Squaring both sides) so $2+2|\vec a||\vec b|\cos θ=1$ $\cos θ=-\frac{1}{2}⇒θ=\frac{2\pi }{3}$