A unit vector \( \vec{a} \) makes angles \( \displaystyle\frac{\pi}{4} \) and \( \displaystyle\frac{\pi}{3} \) with \( x \)-axis and \( y \)-axis respectively and an acute angle \( \theta \) with \( z \)-axis, then the value of \( \theta \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{\pi}{3} \)
The direction cosines of any line satisfy \( l^2+m^2+n^2=1 \):
\( \displaystyle \cos^2\frac{\pi}{4}+\cos^2\frac{\pi}{3}+\cos^2\theta=1 \)
\( \displaystyle \frac{1}{2}+\frac{1}{4}+\cos^2\theta=1 \)
\( \displaystyle \cos^2\theta=\frac{1}{4} \)
Since \( \theta \) is acute, \( \cos\theta=\frac{1}{2} \), so \( \theta=\frac{\pi}{3} \).