Question

Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.

Answer 1

Answer:

Putting y=0 and then y=1, two of the points of the first line are (q, 0, s) and (p+q, 1, r+s), so a vector in the direction of the line is just the difference

(p+q, 1, r+s) - (q, 0, s) = (p, 1, r).

Similarly, a vector in the direction of the second line is

(p', 1, r').

Now using the fact that two vectors are perpendicular if their dot product is zero; that is, if

(p, 1, r) . (p', 1, r') = 0

or expanded

pp' + 1 + rr' = 0.