JEE Advanced 2018 · Paper 1, Physics question 7

Two vectors $\vec A$ and $\vec B$ are defined as $\vec A = a\,\hat i$ and $\vec B = a(\cos\omega t\,\hat i + \sin\omega t\,\hat j)$, where $a$ is a constant and $\omega = \pi/6\ \mathrm{rad\,s^{-1}}$. If $|\vec A + \vec B| = \sqrt3|\vec A - \vec B|$ at time $t = \tau$ for the first time, the value of $\tau$, in seconds, is ______.