If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$ and $|\vec{a} \times \vec{b}| = 3\sqrt{3}$, then the value of $\vec{a} \cdot \vec{b}$ is

3

The correct answer is Option (2) → 3 ##

Given that

$|\vec{a} \times \vec{b}| = 3\sqrt{3}$

$|\vec{a}||\vec{b}| \sin \frac{\pi}{3} = 3\sqrt{3}$

$|\vec{a}||\vec{b}| = 3\sqrt{3} \times \frac{2}{\sqrt{3}} = 6$

$\text{Now, } \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \frac{\pi}{3}$

$= 6 \times \frac{1}{2} = 3$