Question

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 $\omega = \pi/6\, rad\, s^{s-1}. If \left|\vec{A}+\vec{B}\right| = \sqrt{3}\left|\vec{A}-\vec{B}\right|at$ time $t = \tau$ for the first time, the value of $\tau$, in seconds, is __________.

Answer 1

$|\bar{A}+\bar{B}|=2 a \cos \frac{\omega t}{2} $
$|\bar{A}-\bar{B}|=2 a \sin \frac{\omega t}{2}$
So
$2 a \cos \frac{\omega t}{2}=\sqrt{3}\left(2 a \sin \frac{w t}{2}\right) \,\,\, \tan \frac{\omega t}{2}=\frac{1}{\sqrt{3}} $
$ \frac{\omega t}{2}=\frac{\pi}{6} \Rightarrow \omega t=\frac{\pi}{3} $
$ \frac{\pi}{6} t=\frac{\pi}{3} \Rightarrow t=2.00 \,sec$