Question

If vectors $\overrightarrow{A}=\cos wt\hat{i}+\sin wt\hat{j}$ and $\overrightarrow{B}=\cos wt/2\hat{i}+\sin wt/2\hat{j}$ are functions of time , then the value of $t$ at which they are orthogonal to each other is

• A. $t=0$
• B. $t=\frac{\pi }{4\omega }$
• C. $t=\frac{\pi }{2\omega }$
• D. $t=\frac{\pi }{\omega }$

Answer 1

The correct option is D $t=\frac{\pi }{\omega }$

$\overrightarrow{A}.\overrightarrow{B}=(\cos wt\hat{i}+\sin wt\hat{j}).(\cos \frac{wt}{2}\hat{i}+\sin \frac{wt}{2}\hat{j})$
$=\cos (wt-\frac{wt}{2})$
When they are orthogonal $\overrightarrow{A}.\overrightarrow{B}=0$
=> $\cos (wt-\frac{wt}{2})=0$
$\cos (\frac{wt}{2})=0$
$\frac{wt}{2}=\frac{\pi }{2}$
$t=\frac{\pi }{w}$