Question

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

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

Answer 1

Answer: (C)

Given, $a+b+\sqrt{3} c=0$
$\Rightarrow a+b=-\sqrt{3} c\,\,\,...(i)$
$\Rightarrow (a+b) \cdot(a+b)=(-\sqrt{3} c)(-\sqrt{3} c)$
$\Rightarrow | a |^{2}+2 a \cdot b +| b |^{2}=3| c |^{2}$
Since, $| a |,| b |$ and $| c |$ are unit vectors.
$\therefore (1)^{2}+2(1)(1) \cos \theta+1=3(1)$
$\Rightarrow 1+2 \cos \theta+1=3$
$\Rightarrow 2 \cos \theta=3-2$
$\Rightarrow 2 \cos \theta=1$
$\Rightarrow \cos \theta=\frac{1}{2}=\cos \frac{\pi}{3}$
$\Rightarrow \theta=\frac{\pi}{3}$