#### Question

Sum

Find the value of λ for which the following lines are perpendicular to each other `("x"-5)/(5λ+2) = (2 -"y")/(5) = (1 -"z")/(-1); ("x")/(1) = ("y"+1/2)/(2λ) = ("z" -1)/(3)`

hence, find whether the lines intersect or not

#### Solution

`("x"-5)/(5λ+2) = ("y"-2)/(-5) = ("z"-1)/(1)` ....(1)

and

`("x")/(1) = ("y"+1/2)/(2λ) = ("z" -1)/(3)` ...(2)

a_{1 }= 5λ + 2, b_{1} = -5, c_{1} = 1 and

a_{2} = 1, b_{2} = 2λ, c_{2} = 3

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

(5λ + 2) - 5(2λ) + 1(3) = 0

-5λ + 5 = 0

⇒ λ = -1

Concept: Plane - Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point

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