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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.
Solution
Given x = py + q ⇒ y =
And z = ry + s ⇒ y =
So, the equation becomes
Similarly x = p'y + q' ⇒ y =
And z = r'y + s' ⇒ y =
Hence, the equation becomes
If the lines are perpendicular to each other, then
a1a2 + b1b2 + c1c2 = 0
pp' + 1.1 + rr' = 0
Thus, the given lines are perpendicular if pp' + rr' + 1 = 0.
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