Question

If a line makes angles \( 90^\circ, 60^\circ \) and \( \theta \) with \( x, y \) and \( z \) axes respectively, where \( \theta \) is acute, then the value of \( \theta \) is:

• A. \( \dfrac{\pi}{4} \)
• B. \( \dfrac{\pi}{3} \)
• C. \( \dfrac{\pi}{2} \)
• D. \( \dfrac{\pi}{6} \)

Hint:

For any vector in 3D, the sum of the squares of the direction cosines of the angles with the coordinate axes equals 1.

Answer 1

The Correct Option is D

We use the relation: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Given: \[ \cos 90^\circ = 0, \quad \cos 60^\circ = \dfrac{1}{2}, \quad \cos^2 \theta = 1 - 0^2 - \left(\dfrac{1}{2}\right)^2 \] \[ \cos^2 \theta = 1 - \dfrac{1}{4} = \dfrac{3}{4} \] \[ \cos \theta = \dfrac{\sqrt{3}}{2} \] Thus, \( \theta = \dfrac{\pi}{6} \).