Question

If a line \( L \) makes angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) with the positive X-axis and positive Y-axis respectively, then the angle made by \( L \) with the positive direction of the Z-axis is:

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{5\pi}{12} \)

Hint:

Use the relation \( \cos^2 \theta_X + \cos^2 \theta_Y + \cos^2 \theta_Z = 1 \) to find the angle with the Z-axis when the angles with the X and Y axes are known.

Answer 1

The Correct Option is B

Step 1: Use the direction cosine relation.
The direction cosines satisfy: \[ \cos^2 \theta_X + \cos^2 \theta_Y + \cos^2 \theta_Z = 1 \]
Step 2: Substitute the given angles.
We have \( \cos \theta_X = \frac{1}{2} \), \( \cos \theta_Y = \frac{1}{\sqrt{2}} \). Substituting into the equation: \[ \cos^2 \theta_Z = \frac{1}{4} \]
Step 3: Solve for \( \theta_Z \).
\[ \cos \theta_Z = \frac{1}{2}, \quad \theta_Z = \frac{\pi}{3} \] Final Answer: \[ \boxed{\frac{\pi}{3}} \]