Question

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

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

Answer 1

Answer: (A)

$30^\circ$

Answer 2

Given, $| \vec{a} |=| \vec{b} |=|\sqrt{3} \vec{a} - \vec{b} |= 1$
$\Rightarrow |\sqrt{3} \vec{a}-\vec{b}|^{2}=(\sqrt{3} \vec{a}-\vec{b}) \cdot(\sqrt{3} a-b)$
$\Rightarrow 1=3| \vec{a} |^{2}+| \vec{b} |^{2}-2 \sqrt{3}| \vec{a} || \vec{b} | \cos \theta$
$\Rightarrow 1=3+1-2 \sqrt{3} \cos \,\theta$
$\Rightarrow 2 \sqrt{3} \cos\, \theta=3$
$\Rightarrow \cos \,\theta=\frac{\sqrt{3}}{2}$
$\Rightarrow \theta=30^{\circ}$