Question

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

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

Answer 1

Answer: (B)

Let angle between $\hat{a}$ and $\hat{c}$ be $\theta$ .
Now, $\hat{a}-\sqrt{3}\hat{b}+\hat{c} =\overrightarrow{0}$
$\Rightarrow \left(\hat{a}+\hat{c}\right)=\sqrt{3}\hat{b}$
$\Rightarrow \left(\hat{a}+\hat{c}\right)\cdot \left(\hat{a}+\hat{c}\right)=3\left(\hat{b}.\hat{b}\right)$
$\Rightarrow \hat{a.}\hat{a}+\hat{a}.\hat{c}+\hat{c}.\hat{a}+\hat{c}.\hat{c}=3\times1$
$\Rightarrow 1+2\,cos\,\theta+1=3$
$\Rightarrow cos\,\theta =\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}$