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

Answer 1

Answer: 2

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