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.


Given: Lines x = py + q, z = ry + s and x = p’y + q’, z = r’y + s’ are perpendicular.


To Prove: pp’ + rr’ + 1 = 0


Proof: Take x = py + q and z = ry + s


From x = py + q,


py = x – q



From z = ry + s,


ry = z – s



So,



Or,


…(i)


Now, take x = p’y + q’ and z = r’y + s’


From x = p’y + q’,


p’y = x – q’



From z = r’y + s’,


r’y = z – s’



So,



Or,


…(ii)


From (i),


Line L1 is parallel to . [From the denominators of the equation (i)]


From (ii),


Line L2 is parallel to . [From the denominators of the equation (ii)]


According to the question, the lines are perpendicular.


Then, the dot product of the vectors must be equal to 0.


That is,



pp’ + 1 + rr’ = 0


[, by vector dot multiplication, ]


Or,



Thus, the given lines are perpendicular if pp’ + rr’ + 1 = 0.


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