Let a, b, c be three vectors such that | a |=√ 31, 4| b |=| c |=2 and 2( a × b )=3( c × a ). If the angle between b and c is (2 π)/3, then ((a ×…

Let $\displaystyle \vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\displaystyle |\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2$ and $\displaystyle 2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$. If the angle between $\displaystyle \vec{b}$ and $\displaystyle \vec{c}$ is $\displaystyle \frac{2 \pi}{3}$, then $\displaystyle \left(\frac{\vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}}\right)^2$ is equal to $\displaystyle \_\_\_\_$.