Question

If the lines \[ \frac{x - 1}{3} = \frac{y - 2}{2k} = \frac{z - 3}{2} \quad {and} \quad \frac{x - 1}{3k} = \frac{y - 1}{1} = \frac{z - 6}{-5} \] are perpendicular, find the value of \( k \).

Hint:

For two lines to be perpendicular, the dot product of their direction ratios must equal zero.

Answer 1

Step 1: The condition for the perpendicularity of two lines is that the dot product of their direction ratios is zero. The direction ratios of the first line are \( (3, 2k, 2) \), and the direction ratios of the second line are \( (3k, 1, -5) \). 

Step 2: The dot product of these direction ratios is: \[ 3 \cdot 3k + 2k \cdot 1 + 2 \cdot (-5) = 0 \] \[ 9k + 2k - 10 = 0 \] \[ 11k = 10 \] \[ k = \frac{10}{11} \] Thus, the value of \( k \) is \( \frac{10}{11} \).