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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 \( \omega=\frac{\pi}{6} \mathrm{rad} \mathrm{s}^{-1} \). If \( |\vec{A}+\vec{B}|=\sqrt{3}|\vec{A}-\vec{B}| \) at time \( t=\tau \) for the first time, the value of \( \tau \), in seconds, is
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