Answer
Please see the work below.
Work Step by Step
It is given that
$\vec{A}^{\,}.\vec{B}^{\,}=\frac{1}{2}|\vec{A}^{\,}\times \vec{B}^{\,}|$
We know that
$\vec{A}^{\,}.\vec{B}^{\,}=ABcos\theta$
$\vec{A}^{\,}\times \vec{B}^{\,}=ABsin\theta$
$\implies ABcos\theta=\frac{1}{2} ABsin\theta$
$2=tan\theta$
$\theta=tan^{-1}(2)=63.4^{\circ}$
