Question

Write down a unit vector In $XY-$Plane, Making an angle of $30$ degrees with the positive direction of $x-$Axis.

Answer 1

Unit vector.

A unit vector $\overrightarrow{r}$in $XY-$plane is written as,

$\overrightarrow{r}=\cos \theta \hat{i}+\sin \theta \hat{j}$

Where $\theta$ is the angle made by the unit vector with the positive direction of the $x-$axis.

Given that, $\theta =30°$

$$\begin{gathered} \Rightarrow \overrightarrow{r}=\cos 30°\hat{i}+\sin 30°\hat{j} \\ \Rightarrow \overrightarrow{r}=\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{30}\mathbf{°}\mathbf{=}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{30}\mathbf{°}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{]} \\ \therefore \overrightarrow{r}=\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j} \end{gathered}$$

Hence, the required unit vector is $\overrightarrow{r}=\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j}$.