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 angles \( \alpha, \beta, \gamma \) made by the line with the \( x \)-axis, \( y \)-axis, and \( z \)-axis respectively, satisfy the equation for direction cosines: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1. \] Given that \( \alpha = \frac{\pi}{4} \) and \( \gamma = \frac{\pi}{4} \), we calculate: \[ \cos \alpha = \cos \gamma = \frac{\sqrt{2}}{2}. \] Substitute these values into the equation: \[ \left(\frac{\sqrt{2}}{2}\right)^2 + \cos^2 \beta + \left(\frac{\sqrt{2}}{2}\right)^2 = 1. \] Simplify: \[ \frac{1}{2} + \cos^2 \beta + \frac{1}{2} = 1 \quad \Rightarrow \quad \cos^2 \beta = 0. \] This implies: \[ \cos \beta = 0 \quad \Rightarrow \quad \beta = \frac{\pi}{2}. \]
Final Answer: \( \boxed{\frac{\pi}{2}} \).