Vectors through products.

Let $\vec a, \vec b, \vec c$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac {\pi}{3}$ . If ${\vec a} \times {\vec b} + {\vec b} \times {\vec c} = p \vec a + q \vec b + r \vec c$ , where $p, q, r$ are scalars, then evaluate

$\large \frac {p^2 + 2q^2 + r^2}{q^2}$ .