Let L 1 and L 2 be the lines whose equation are 
and 
= 
= 
respectively. A and B are two points on L 1 and L 2 respectively such that AB is perpendicular both the lines L 1 and L 2 .Find points A, B and hence find shortest distance between lines L 1 and L 2
Text Solution
Verified by ExpertsA
A = (3, 8, 3), B = (–3, – 7, 6), AB = 3 
Sol. Let the co-ordinates of A be (3 λ + 3, 8 – λ , λ + 3) and the co-ordinates of B be (–3 μ – 3, 2 μ – 7, 4 μ + 6).
Then direction ratios of AB are 3 λ + 3 μ + 6, – λ – 2 μ + 15, λ – 4 μ – 3
AB ⊥ L 1 so 3(3 λ + 3 μ + 6) – (– λ – 2 μ + 15) + ( λ – 4 μ – 3) = 0
i.e. 11 λ + 7 μ = 0
and AB ⊥ L 2 so – 3(3 λ + 3 μ + 6) + 2(– λ – 2 μ + 15) + 4( λ – 4 μ – 3) = 0 i.e. –7 λ – 29 μ = 0
⇒ λ = μ = 0 so the point A is (3, 8, 3) and the point B is (–3, –7, 6)
∴ AB = 
= 
= 3 
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