Question

If the direction cosines of two lines are given by \[ l + m + n = 0 \quad \text{and} \quad mn - 2lm - 2nl = 0, \] then the acute angle between those lines is:

• A. \( \frac{2\pi}{5} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{60} \)

Hint:

Use trigonometric properties to solve for angles when given direction cosine relations.

Answer 1

The Correct Option is B

Step 1: Solve for direction cosines Given: \[ l + m + n = 0. \] We solve using the second equation: \[ mn
- 2lm
- 2nl = 0. \] Substituting values, solving for angles, we find: \[ \cos \theta = \frac{1}{2}. \] Step 2: Compute the angle \[ \theta = \cos^{
-1} \frac{1}{2} = \frac{\pi}{3}. \]