Question

The angle made by the vector \(\sqrt{3} \hat{i}+\hat{j}\) with \(x\)-axis is

(a) \(30^{\circ}\)

(b) \(45^{\circ}\)

(c) \(60^{\circ}\)

(d) \(90^{\circ}\)

Answer 1

The correct option of this question will be (a).

Solution —

Let $\theta$ be the angle made by $\sqrt{3} \hat{i}+\hat{j}$ with $x$-axis.

If $\hat{i}$ is the unit vector along $x$-axis, then

$\cos \theta=\frac{(\sqrt{3} \hat{i}+\hat{j}) \cdot \hat{i}}{|\sqrt{3} \hat{i}+\hat{j}||\hat{i}|}$

=$\frac{\sqrt{3}}{\sqrt{\sqrt{3}^{2}+1^{2}} \sqrt{1^{2}}}$

=$\frac{\sqrt{3}}{\sqrt{4}}=\frac{\sqrt{3}}{2}$

or

$\theta=\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=30^{\circ}$