Find p If Two Lines Are Perpendicular | 3D Geometry Solution

 If the lines (x - 3)/1 = (1 - y)/1 = (z + 2)/p and (2 - x)/3 = (y + 1)/5 = (z + 56)/2p are perpendicular to each other, then find the value(s) of p.

For a line in symmetric form (x - x1)/a = (y - y1)/b = (z - z1)/c, 

the direction ratios are (a, b, c).

first line

(x - 3)/1 = (1 - y)/1 = (z + 2)/p

rewrite

 1 - y as -(y - 1) or (y - 1)/(-1)

So direction ratios first line = (1, -1, p)

for second line  (2 - x)/3 = (y + 1)/5 = (z + 56)/2p

Rewrite 2 - x as -(x - 2) so that the first expression changes to (x - 2)/(-3)

So direction ratios of the second line are  (-3, 5, 2p)

Two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) are perpendicular if:

a₁×a₂ + b₁×b₂ + c₁×c₂ = 0

(1)(-3) + (-1)(5) + (p)(2p) = 0

-3 - 5 + 2p² = 0

-8 + 2p² = 0

2p² = 8

p² = 4

p = +2 or p = -2

see this video for more explanation