Condition for lines to be perpendicular

Two lines are perpendicular if the dot product of their direction ratios is zero.

Direction ratios

The line x = ay + b, z = cy + d can be written in symmetric form by solving for y:

(x − b)/a = y/1 = (z − d)/c.

So its direction ratios are (a, 1, c). (Equivalently, each coordinate is a function of y: x = ay + b, z = cy + d, so for a change of 1 in y, x changes by a and z by c.)

Similarly the second line x = a′y + b′, z = c′y + d′ has direction ratios (a′, 1, c′).

Condition

The lines are perpendicular when

a·a′ + 1·1 + c·c′ = 0

Hence the lines are perpendicular if aa′ + cc′ + 1 = 0.

Example: x = 2y, z = y and x = −y, z = −y: aa′ + cc′ + 1 = −2 − 1 + 1 = −2 ≠ 0, not perpendicular; but x = y, z = y and x = −y, z = 0·y gives (1)(−1) + (1)(0) + 1 = 0, perpendicular.