Question

If a line makes angles of \( \frac{3\pi}{4} \) and \( \frac{\pi}{3} \) with the positive directions of \( x \), \( y \), and \( z \)-axes respectively, then \( \theta \) is:

• A. \( -\frac{\pi}{3} \)
• B. \( \frac{\pi}{3} \) only
• C. \( \frac{\pi}{6} \)
• D. \( \pm \frac{\pi}{3} \)

Hint:

The direction cosines of a line are always in the range \( -1 \) to \( 1 \), and the angle can have multiple solutions due to the symmetry of the coordinate axes.

Answer 1

The Correct Option is D

For a line making angles \( \alpha \), \( \beta \), and \( \gamma \) with the positive directions of the \( x \), \( y \), and \( z \)-axes respectively, the direction cosines of the line are:
\[ \cos \alpha = \frac{1}{\sqrt{1^2 + 1^2 + 1^2}}, \quad \cos \beta = \frac{1}{\sqrt{1^2 + 1^2 + 1^2}}, \quad \cos \gamma = \frac{1}{\sqrt{1^2 + 1^2 + 1^2}}. \] The angle \( \theta \) is determined by the geometry of the line and can take both positive and negative values based on the orientations of the line with respect to the axes. Thus, the correct answer is \( \pm \frac{\pi}{3} \).