Question

Write down a unit vector in $XY$- plane, making an angle of ${30}^{\circ }$ with the positive direction of $x-axis$ .

Answer 1

If $\overrightarrow{r}$ is a unit vector in the $XY-plane$, then $\overrightarrow{r}=\cos \theta \hat{i}+\sin \theta \hat{j}$.
Here, $\theta$ is the angle made by the unit vector with the positive direction of the $x-axis$
Therefore, for $\theta ={30}^{\circ }$:
$\overrightarrow{r}=\cos {30}^{\circ }+\sin {30}^{\circ }\hat{j}=\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j}$
Hence, the required unit vector is $\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j}$.