For non-zero vectors $\vec{a}$ and $\vec{b}$, $\lvert \vec{a} \times \vec{b} \rvert = \sqrt{3}\,(\vec{a} \cdot \vec{b})$. What is the angle between them?

Explanation

1. 01

   $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin \theta = \sqrt{3}\lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos \theta$, so $\tan \theta = \sqrt{3}$.

2. 02

   With $\theta$ in $[0, \pi]$ and $\cos \theta > 0$, $\theta = \frac{\pi}{3}$.