Question

Find the value of \(k\) if the lines \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x-4}{k} = \frac{y-1}{2} = \frac{z}{1} \) are perpendicular.

• A. \(5\)
• B. \(-5\)
• C. \(2\)
• D. \(-2\)

Hint:

If two lines in 3D are perpendicular, the dot product of their direction ratios must be zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem states that two lines in 3D space are perpendicular. We need to find an unknown parameter \( k \) in the direction ratios of the second line.
Step 2: Key Formula or Approach:
Two lines with direction ratios \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) are perpendicular if and only if their dot product is zero:
\[ a_1a_2 + b_1b_2 + c_1c_2 = 0 \]
Step 3: Detailed Explanation:
From the equations of the lines:
Line 1: \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \)
The direction ratios are \( a_1 = 2, b_1 = 3, c_1 = 4 \).
Line 2: \( \frac{x-4}{k} = \frac{y-1}{2} = \frac{z}{1} \)
The direction ratios are \( a_2 = k, b_2 = 2, c_2 = 1 \).
Applying the perpendicularity condition:
\[ (2)(k) + (3)(2) + (4)(1) = 0 \]
\[ 2k + 6 + 4 = 0 \]
\[ 2k + 10 = 0 \]
\[ 2k = -10 \]
\[ k = -5 \]
Step 4: Final Answer:
The value of \( k \) is \( -5 \).