Question

If $\overrightarrow{a}\&\overrightarrow{b}$ are unit vectors, then angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ for $\sqrt{3}\overrightarrow{a}-\overrightarrow{b}$ to be unit vector is

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is A ${30}^{\circ }$

Let $\theta$ be the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
Given that $\sqrt{3}\overrightarrow{a}-\overrightarrow{b}$ is a unit vector
So, $|\sqrt{3}\overrightarrow{a}-\overrightarrow{b}|=1$

$\Rightarrow 3|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2\cdot \sqrt{3}\overrightarrow{a}\cdot \overrightarrow{b}=1$

$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=\frac{\sqrt{3}}{2}$

$\Rightarrow \cos \theta =\frac{\sqrt{3}}{2}$

$\Rightarrow \theta ={30}^{\circ }$