Q96 / 180 · KCET 2016 · Mathematics

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?

  • Correct answer
    A
    \( 30^{\circ} \)
  • B
    \( 45^{\circ} \)
  • C
    \( 60^{\circ} \)
  • D
    \( 90^{\circ} \)

Solution

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

Hence, the correct answer is option: 'A'.

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