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$\vec{A}$ and $\vec{B}$ are two vectors and $\theta$ is the angle between them, if $|\vec{A} \times \vec{B}| = \sqrt{3}\,(\vec{A}\cdot\vec{B})$, the value of $\theta$ is
A
$45^\circ$
B
$30^\circ$
C
$90^\circ$
D
$60^\circ$
Detailed Solution
$|\vec{A} \times \vec{B}| = AB\sin\theta$ and $\vec{A}\cdot\vec{B} = AB\cos\theta$
Given $|\vec{A} \times \vec{B}| = \sqrt{3}\,(\vec{A}\cdot\vec{B})$
$\Rightarrow AB\sin\theta = \sqrt{3}\,AB\cos\theta$
$\Rightarrow \tan\theta = \sqrt{3}$
$\Rightarrow \theta = 60^\circ$
Given $|\vec{A} \times \vec{B}| = \sqrt{3}\,(\vec{A}\cdot\vec{B})$
$\Rightarrow AB\sin\theta = \sqrt{3}\,AB\cos\theta$
$\Rightarrow \tan\theta = \sqrt{3}$
$\Rightarrow \theta = 60^\circ$