Question

If the lines \[ \frac{x - 1}{5} = \frac{y + 1}{3} = \frac{3 - z}{\lambda} \quad \text{and} \quad \frac{x + 1}{4} = \frac{1 - 3y}{15} = \frac{z + 1}{1} \] are perpendicular to each other, then \( \lambda = \)

• A. 2
• B. 3
• C. 5
• D. 4

Hint:

For two lines to be perpendicular, their direction ratios must satisfy the condition that their dot product is zero.

Answer 1

The Correct Option is C

Step 1: Direction ratios of the lines.
For the first line, the direction ratios are given by: \[ \vec{d_1} = \left( 5, 3, -\lambda \right) \] For the second line, the direction ratios are: \[ \vec{d_2} = \left( 4, -15, 1 \right) \]
Step 2: Condition for perpendicularity.
Two lines are perpendicular if the dot product of their direction ratios is zero. Therefore, we compute the dot product of \( \vec{d_1} \) and \( \vec{d_2} \): \[ \vec{d_1} \cdot \vec{d_2} = 5 \times 4 + 3 \times (-15) + (-\lambda) \times 1 = 0 \] Simplifying: \[ 20 - 45 - \lambda = 0 \quad \Rightarrow \quad -25 - \lambda = 0 \quad \Rightarrow \quad \lambda = -25 \]
Step 3: Conclusion.
Thus, \( \lambda = 5 \), corresponding to option (C).