Question

The direction cosines of the line making angles $$ \frac{\pi}{4}, \frac{\pi}{3} $$ and $ \theta $ (where $ 0<\theta<\frac{\pi}{2} $) with X, Y, and Z axes respectively are:

• A. \( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2} \)
• B. \( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{\sqrt{3}}{2} \)
• C. \( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{\sqrt{2}} \)
• D. None of these

Hint:

Sum of squares of direction cosines equals 1.

Answer 1

The Correct Option is A

Since direction cosines \( l = \cos \alpha, m = \cos \beta, n = \cos \gamma \), with \[ \alpha = \frac{\pi}{4}, \quad \beta = \frac{\pi}{3}, \quad \gamma = \theta \] Using relation: \[ l^2 + m^2 + n^2 = 1 \] \[ \Rightarrow \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Substitute: \[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2 \gamma = 1 \] \[ \Rightarrow \frac{1}{2} + \frac{1}{4} + \cos^2 \gamma = 1 \] \[ \Rightarrow \cos^2 \gamma = \frac{1}{4} \Rightarrow \cos \gamma = \frac{1}{2} \] Therefore, direction cosines are: \[ \left( \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2} \right) \]