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\( \mathrm{P} \)
Two vectors \( \overrightarrow{\mathrm{A}} \) and \( \overrightarrow{\mathrm{B}} \) are defined as \( \overrightarrow{\mathrm{A}}=a \hat{i} \) and
W \( \overrightarrow{\mathrm{B}}=\mathrm{a}(\cos \omega t \hat{\mathrm{i}}+\sin \omega \hat{\mathrm{j}}) \), where a is a constant and \( \omega=\pi / 6 \mathrm{rad} \mathrm{s}^{-1} \). If \( |\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}|=\sqrt{3}|\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}| \) at time \( \mathrm{t}=\tau \) for the first time, the value of \( \tau \), in seconds, is
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