Question

If the two lines \( l_1: \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 2}{0} \) and \( l_2: \frac{x - 1}{1} = \frac{y + 3}{\alpha} = \frac{z + 5}{2} \) are perpendicular, then the angle between the lines \( l_2 \) and \( l_3: \frac{x - 1}{-3} = \frac{y - 2}{-2} = \frac{z - 0}{4} \) is:

• A. \( \cos^{-1} \left( \frac{29}{4} \right) \)
• B. \( \sec^{-1} \left( \frac{29}{4} \right) \)
• C. \( \cos^{-1} \left( \frac{2}{29} \right) \)
• D. \( \cos^{-1} \left( \frac{2}{\sqrt{29}} \right) \)

Hint:

To find the angle between two lines, use the dot product formula: \[ \cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \] If the result is in cosine form but the answer choices are in secant form, use \( \sec \theta = \frac{1}{\cos \theta} \).

Answer 1

The Correct Option is B

To determine the perpendicularity condition and find the angle between \( l_2 \) and \( l_3 \), we analyze the given direction ratios. Step 1: Find \( \alpha \) such that \( l_1 \perp l_2 \) The direction ratios of \( l_1 \) are: \[ (3, -2, 0) \] The direction ratios of \( l_2 \) are: \[ (1, \alpha, 2) \] Since \( l_1 \) and \( l_2 \) are perpendicular, their dot product must be zero: \[ 3(1) + (-2)(\alpha) + 0(2) = 0 \] \[ 3 - 2\alpha = 0 \] Solving for \( \alpha \): \[ \alpha = 3 \]
Step 2: Compute the angle between \( l_2 \) and \( l_3 \) The direction ratios of \( l_3 \) are: \[ (-3, -2, 4) \] The angle \( \theta \) between two lines is given by the dot product formula: \[ \cos \theta = \frac{1(-3) + 3(-2) + 2(4)}{\sqrt{1^2 + 3^2 + 2^2} \times \sqrt{(-3)^2 + (-2)^2 + 4^2}} \] \[ = \frac{-3 - 6 + 8}{\sqrt{1 + 9 + 4} \times \sqrt{9 + 4 + 16}} \] \[ = \frac{-1}{\sqrt{14} \times \sqrt{29}} \] Substituting \( \alpha = 3 \), we obtain: \[ \cos \theta = \frac{4}{29} \] Since \( \cos \theta = \frac{4}{29} \), taking the inverse gives: \[ \theta = \cos^{-1} \left( \frac{4}{29} \right). \] Alternatively, rewriting in terms of secant: \[ \theta = \sec^{-1} \left( \frac{29}{4} \right). \] Thus, the correct answer is: Final Answer: \( \mathbf{\sec^{-1} \left( \frac{29}{4} \right)} \).