Question

If a line makes an angle of \( \frac{\pi}{4} \) with the positive directions of both \( x \)-axis and \( z \)-axis, then the angle which it makes with the positive direction of \( y \)-axis is:

• A. \( 0 \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{2} \)
• D. \( \pi \)

Hint:

The direction cosines of a line satisfy \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \). Use known angles to compute the unknown.

Answer 1

The Correct Option is C

The direction cosines of a line with respect to the \( x \), \( y \), and \( z \) axes are denoted by \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \) respectively. They satisfy the identity \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \). Given \( \alpha = \frac{\pi}{4} \) and \( \gamma = \frac{\pi}{4} \), we have \( \cos \alpha = \cos \gamma = \frac{\sqrt{2}}{2} \). Substituting these into the identity yields \( \left(\frac{\sqrt{2}}{2}\right)^2 + \cos^2 \beta + \left(\frac{\sqrt{2}}{2}\right)^2 = 1 \). This simplifies to \( \frac{1}{2} + \cos^2 \beta + \frac{1}{2} = 1 \), which further reduces to \( \cos^2 \beta = 0 \). Therefore, \( \cos \beta = 0 \), implying \( \beta = \frac{\pi}{2} \).
Final Answer: \( \boxed{\frac{\pi}{2}} \).