Question

If θ is an acute angle and the vector (sin θ)$\hat{i}$ + (cos θ)$\hat{j}$ is perpendicular to the vector $\hat{i}-\sqrt{3}\hat{j},$ then θ =
(a) $\frac{\mathrm{\pi }}{6}$

(b) $\frac{\mathrm{\pi }}{5}$

(c) $\frac{\mathrm{\pi }}{4}$

(d) $\frac{\mathrm{\pi }}{3}$

Answer 1

(d) $\frac{\mathrm{\pi }}{3}$

$$\begin{gathered} \text{The given vectors are perpendicular. So, their dot product is zero.} \\ \left[\left(\sin \theta \right)\hat{i}+\left(\cos \theta \right)\hat{j}\right].\left(\hat{i}-\sqrt{3}\hat{j}\right)=0 \\ \Rightarrow \sin \theta -\sqrt{3}\cos \theta =0 \\ \Rightarrow \sin \theta =\sqrt{3}\cos \theta \\ \Rightarrow \tan \theta =\sqrt{3} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3}\text{(Because θ is acute)} \end{gathered}$$