# If the sum of two unit vector is a unit vector, then what is the angle between them?

$\begin{array}{1 1} \large\frac{\pi }{6} \\\large\frac{\pi }{3} \\ \large\frac{ 2 \pi }{3}\\ \large\frac{\pi }{8} \end{array}$

Toolbox:
• $|\overrightarrow a+\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+2\overrightarrow a.\overrightarrow b$
• $\overrightarrow a.\overrightarrow b=|\overrightarrow a|\:|\overrightarrow b|\:cos\theta$
Given: $\overrightarrow a,\overrightarrow b\:\:and\:\:\overrightarrow a+\overrightarrow b$ are unit vectors.
$\Rightarrow\:|\overrightarrow a|=|\overrightarrow b|=|\overrightarrow a+\overrightarrow b|=1$
We know that $|\overrightarrow a+\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+2\overrightarrow a.\overrightarrow b$
$\Rightarrow\:|\overrightarrow a+\overrightarrow b|^2=|\overrightarrow a|^2+|\overrightarrow b|^2+2|\overrightarrow a|\:|\overrightarrow b|\:cos\theta$
$\Rightarrow\:1=1+1+2cos\theta=2+2cos\theta$
$\Rightarrow\:cos\theta=-\frac{1}{2}$
$\therefore\:\theta=\large\frac{2\pi}{3}$