If the lines x=ay+b, z=cy+d and x=a'z+b', y=c'z+d' are perpendicular, then:

Line x=ay+b, z=cy+d ⇒ x-b/a = y = (z-d)/c
Line x=a'z+b', y=c'z+d' ⇒ (x-b')/a' = (y-d')/c' = z
Given both lines are perpendicular ⇒ aa'+cc'+c=0