Question

If the direction ratios of two lines are given by \( 3lm - 4ln + mn = 0 \) and \( l + 2m + 3n = 0 \), then the angle between the lines is:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

To find the angle between two lines, use the formula involving the direction cosines of the two lines. The direction cosines can be derived from the direction ratios after solving the given system of equations.

Answer 1

The Correct Option is D

The direction ratios of the two lines are given as: 1. \( 3lm - 4ln + mn = 0 \) 2. \( l + 2m + 3n = 0 \) To find the angle between the lines, we first calculate the direction cosines of the two lines using the given direction ratios. We are provided with two equations that represent the direction ratios of the two lines. Solving this system of equations can give us the direction ratios, from which we can find the angle between the two lines. The formula for the angle \( \theta \) between two lines with direction cosines \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) is: \[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}} \] By solving the system of equations and applying the formula for the angle between the lines, we find that the angle between the lines is \( \frac{\pi}{2} \). Therefore, the correct answer is option (D)