If line 2 x - 4 λ = y - 1 2 = z - 3 1 and x - 1 1 = 3 y - 1 λ = z - 2 1 are perpendicular to each other then λ = …

Step-by-step solution

Key Idea If lines x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 and x - x 2 a 2 = y - y 2 b 2 = z - z 2 c 2 are perpendicular, then a 1 a 2 + b 1 b 2 + c 1 c 2 = 0 Given lines are 2 x - 4 λ = y - 1 2 = z - 3 1 ⇒ x - 2 λ 2 = y - 1 2 = z - 3 1 … i and x - 1 1 = 3 y - 1 λ = z - 2 1 ⇒ x - 1 1 = y - 1 3 λ 3 = z - 2 1 … i i Since, lines (i) and (ii) are perpendicular. ∴ λ 2 × 1 + 2 × λ 3 + 1 × 1 = 0 ⇒ λ 2 + 2 λ 3 + 1 = 0 ⇒ 3 λ + 4 λ + 6 = 0 ⇒ 7 λ = - 6 ⇒ λ = - 6 7