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:

For lines in 3D geometry, the sum of the squares of the direction cosines always equals 1.

Answer 1

The Correct Option is C

Step 1: Recall the direction cosine condition
For a line making angles \( \alpha, \beta, \gamma \) with the positive directions of the \( x \)-axis, \( y \)-axis, and \( z \)-axis respectively, the sum of the squares of the direction cosines is: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1. \] 
Step 2: Substitute the given angles
The line makes angles \( \alpha = \frac{\pi}{4} \) and \( \gamma = \frac{\pi}{4} \) with the \( x \)-axis and \( z \)-axis, so: \[ \cos^2 \frac{\pi}{4} + \cos^2 \beta + \cos^2 \frac{\pi}{4} = 1. \] Since \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \), we have: \[ \left( \frac{1}{\sqrt{2}} \right)^2 + \cos^2 \beta + \left( \frac{1}{\sqrt{2}} \right)^2 = 1. \] Simplify: \[ \frac{1}{2} + \cos^2 \beta + \frac{1}{2} = 1. \] 
Step 3: Solve for \( \cos^2 \beta \)
Combine terms: 
\[ 1 + \cos^2 \beta = 1 \implies \cos^2 \beta = 0. \] Thus: \[ \cos \beta = 0 \implies \beta = \frac{\pi}{2}. \] 
Step 4: {Conclude the result}
The angle which the line makes with the positive direction of the \( y \)-axis is \( \frac{\pi}{2} \).