If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three unit vectors, then prove that:
\[
|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 \leq 9
\]
Show Hint
This expression achieves its absolute maximum value of 9 when the three vectors point in symmetrically opposing directions in a plane ($120^\circ$ angles from each other), causing their mutual dot products to equal $-\frac{1}{2}$.