Question

Find the value of \(p\) so that the lines \[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z+17}{p} \quad \text{and} \quad \frac{x+4}{2} = \frac{y+9}{2} = \frac{z-1}{2} \] are mutually perpendicular.

Hint:

Lines are perpendicular if the dot product of their direction vectors is zero.

Answer 1

Direction ratios of the first line: \[ (2, 3, p). \] Direction ratios of the second line: \[ (2, 2, 2). \] For the lines to be mutually perpendicular, their direction vectors must satisfy: \[ 2 \times 2 + 3 \times 2 + p \times 2 = 0, \] \[ 4 + 6 + 2p = 0, \] \[ 10 + 2p = 0 \implies 2p = -10 \implies p = -5. \]
Final answer: \[ \boxed{p = -5}. \]