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:
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:
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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.
Updated On: Mar 18, 2026
- \( \frac{\pi}{2} \)
- \( \frac{\pi}{3} \)
- \( \frac{\pi}{4} \)
- \( \frac{5\pi}{12} \)
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The Correct Option is B
Solution and Explanation
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: \[ \boxed{\frac{\pi}{3}} \]
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: \[ \boxed{\frac{\pi}{3}} \]
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