If θ is an acute angle and the vector $\vec a = (\sin θ)\hat i + (\cos θ)\hat j$ is perpendicular to the vector $\vec b=\hat i-\sqrt{3}\hat j$ then θ is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\frac{\pi}{3}$
Given $\vec a=(\sin\theta,\cos\theta)$ and $\vec b=(1,-\sqrt{3})$.
Since vectors are perpendicular:
$\vec a\cdot \vec b=0$
$\sin\theta\cdot 1+\cos\theta(-\sqrt{3})=0$
$\sin\theta-\sqrt{3}\cos\theta=0$
$\tan\theta=\sqrt{3}$
Since $\theta$ is acute:
$\theta=\frac{\pi}{3}$
Final answer: $\frac{\pi}{3}$