If \(\theta\) is an acute angle and the vector \(\vec a = (\sin\theta)\hat{\imath} + (\cos\theta)\hat{\jmath}\) is perpendicular to the vector \(\vec b = \hat{\imath} - \sqrt{3}\,\hat{\jmath}\) then \(\theta\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{3}\)
Two vectors are perpendicular when their dot product is zero:
\(\vec a \cdot \vec b = \sin\theta - \sqrt{3}\cos\theta = 0\)
\(\tan\theta = \sqrt{3}\)
Since \(\theta\) is acute:
\( \displaystyle \theta = \frac{\pi}{3} \)