Question

If $\overline{a},\overline{b},\overline{c}$ are unit vectors satisfying the relation $\overline{a}+\overline{b}+\sqrt{3}\overline{c}=0$, then the angle between $\overline{a}$ and $\overline{b}$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C $\frac{\pi }{3}$

$\overrightarrow{a}+\overrightarrow{b}=-\sqrt{3}\overrightarrow{c}$

Sqquaring above equation, we get

$|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2+2|\overrightarrow{a}||\overrightarrow{b}|cos\left(\theta \right)=3|\overrightarrow{c}{|}^2$

$⟹1+1+2\left(1\right)\left(1\right)cos\left(\theta \right)=3\left(1\right)$ ...$(|a|=|b|=|c|=1,unitvectors)$

$⟹cos\left(\theta \right)=\frac{1}{2}$

$⟹\theta =\frac{\pi }{3}$

Hence the answer is option (C)