Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors then what is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ for $\sqrt{3}\overrightarrow{a}-\overrightarrow{b}$ to be unit vecotr?

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

Answer 1

Answer: (A)

${30}^o$

We have been given that, $|\sqrt{3}\overrightarrow{a}-\overrightarrow{b}|=1$

Now square both sides,

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

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

$\Rightarrow 3+1-2\sqrt{3}\cos \theta =1$

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

$\therefore \theta ={30}^o$