Question

Find the direction cosines of two lines which are connected by the relations l + m + n = 0 and mn – 2nl – 2lm = 0.

Answer 1

Given l + m + n = 0 ——— (1)

mn – 2nl – 2lm = 0 ——— (2)

From (1), l = –(m + n)

Substituting in (2),

mn + 2n(m + n) + 2m(m + n) = 0

mn + 2mn + 2n² +2m² + 2mn = 0

2m² + 5mn + 2n² = 0

(2m + n) (m + 2n) = 0

2m = –n or m = –2n

Case (i) : 2m₁ = –n₁

From l₁ = –m₁ – n₁

= –m₁ + 2m₁ = m₁

l₁/1 = m₁/1 = x₁/-2

D.Rs of the first line are 1, 1, –2

D.Cs of this line are 1/√6,1/√6, - 2/√6

Case (ii) : m₂ = –2n₂

From (1) l₂ = –m₂ – n₂ = +2n₂ – n₂ = n₂ 

l₂/1 = m₂/-2 = n₂/1

d.cs of the second line are 1, –2, 1

d.cs of this line are 1/√6, -2/√6, 1/√6