If $\hat{a}$ is a unit vector in the $x y$-plane making an angle $30^{\circ}$ with the positive $x$-axis, then what is $\hat{a}$ equal to ?

• Correct answer
  A

  $\frac{\sqrt{3} \hat{i}+\hat{j}}{2}$
• B

  $\frac{\sqrt{3} \hat{i}-\hat{j}}{2}$
• C

  $\frac{\hat{i}+\sqrt{3} \hat{j}}{2}$
• D

  $\frac{\hat{i}-\sqrt{3} \hat{j}}{2}$

Solution

Let $A(x \hat{i}, y \hat{j})$ is a point in the $x y$-plane. From question, $\angle A O B=30^{\circ} \text { and } \overrightarrow{O A}=1$ $\begin{aligned} & x=|\overrightarrow{O A}| \cdot \cos 30^{\circ}=1 \times \frac{\sqrt{3}}{2} \hat{i}=\frac{\sqrt{3}}{2} \hat{i} \\ & y=|\overrightarrow{O A}| \cdot \sin 30^{\circ}=1 \times \frac{1}{2} \hat{j}=\frac{\hat{j}}{2}\end{aligned}$ $\therefore \vec{a}=\frac{\sqrt{3}}{2} \hat{i}+\frac{1}{2} \hat{j}=\frac{\sqrt{3} \hat{i}+1 \hat{j}}{2}$

Hence, the correct answer is option: 'A'.