Question

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

Answer 1

Solution:

If the lines are perpendicular than their sum of multiplication of direction ratios are zero.

$\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$

Direction ratio of line a1,b1,c1=-3,2k,2

$\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$

Direction ratio of line a2,b2,c2=3k,1,-5

If both the lines are perpendicular

$a1a2 + b1b2 + c1c2 = 0 \\ \\ - 3(3k) + 2k(1) + 2( - 5) = 0 \\ \\ - 9k + 2k - 10 = 0 \\ \\ - 7k = 10 \\ \\ k = \frac{ - 10}{7}$