This question is similar to Chapter 11 Class 12 Three Dimensional Geometry - Ex 11.2

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https://www.teachoo.com/3516/753/Ex-11.2--14---Find-shortest-distance-between-lines/category/Ex-11.2/

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Question 35 (A) Find the shortest distance between the lines 𝑙_1 and 𝑙_2 whose vector equations are π‘Ÿ βƒ—=(βˆ’Δ± Λ†βˆ’Θ· Λ†βˆ’π‘˜ Λ†)+πœ†(7Δ± Λ†βˆ’6Θ· Λ†+π‘˜ Λ†)" and " π‘Ÿ βƒ—=(3Δ± Λ†+5Θ· Λ†+7π‘˜ Λ†)+πœ‡(Δ± Λ†βˆ’2Θ· Λ†+π‘˜ Λ†) where πœ† and πœ‡ are parameters.Shortest distance between the lines with vector equations π‘Ÿ βƒ— = (π‘Ž1) βƒ— + πœ† (𝑏1) βƒ—and π‘Ÿ βƒ— = (π‘Ž2) βƒ— + πœ‡(𝑏2) βƒ— is |(((π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— ).((π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ— ))/|(π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— | | 𝒓 βƒ— = (β€“π’Š Μ‚ – 𝒋 Μ‚ – π’Œ Μ‚) + πœ†(7π’Š Μ‚ βˆ’ 6𝒋 Μ‚ + π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž1) βƒ— + πœ† (𝑏1) βƒ—, (π‘Ž1) βƒ— = 𝑖 Μ‚ – 𝑗 Μ‚ – π‘˜ Μ‚ & (𝑏1) βƒ— = 7𝑖 Μ‚ – 6𝑗 Μ‚ + π‘˜ Μ‚ 𝒓 βƒ— = (3π’Š Μ‚ + 5𝒋 Μ‚ + 7π’Œ Μ‚) + 𝝁 (π’Š Μ‚ – 2𝒋 Μ‚ + π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž2) βƒ— + πœ‡(𝑏2) βƒ— , (π‘Ž2) βƒ— = 3𝑖 Μ‚ + 5𝑗 Μ‚ + 7π‘˜ Μ‚ & (𝑏2) βƒ— = 𝑖 Μ‚ – 2𝑗 Μ‚ + π‘˜ Μ‚ Now, (π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ— = (3𝑖 Μ‚ + 5𝑗 Μ‚ + 7π‘˜ Μ‚) βˆ’ (–𝑖 Μ‚ – 𝑗 Μ‚ – π‘˜ Μ‚) = (3 + 1) 𝑖 Μ‚ + (5 + 1)𝑗 Μ‚ + (7 + 1) π‘˜ Μ‚ = 4π’Š Μ‚ + 6𝒋 Μ‚ + 8π’Œ Μ‚ (π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@7& βˆ’6&1@1&βˆ’2&1)| = 𝑖 Μ‚ [(βˆ’6 Γ— 1)βˆ’(βˆ’2Γ—1)] βˆ’ 𝑗 Μ‚ [(7Γ—1)βˆ’(1Γ—1)] + π‘˜ Μ‚ [(7Γ—βˆ’2)βˆ’(1Γ—βˆ’6)] = 𝑖 Μ‚ [βˆ’6+2] βˆ’ 𝑗 Μ‚ [7βˆ’1] + π‘˜ Μ‚ [βˆ’14+6] = βˆ’4π’Š Μ‚ βˆ’ 6𝒋 Μ‚ – 8π’Œ Μ‚ Magnitude of ((𝑏1) βƒ— Γ— (𝑏2) βƒ—) = √((βˆ’4)2+(βˆ’6)2+(βˆ’8)2) |(π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— | = √(16+36+64) = βˆšπŸπŸπŸ” Also, ((π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ—) . ((π’‚πŸ) βƒ— – (π’‚πŸ) βƒ—) = ("βˆ’4" 𝑖 Μ‚" βˆ’ 6" 𝑗 Μ‚" – 8" π‘˜ Μ‚).(4𝑖 Μ‚ + 6𝑗 Μ‚ + 8π‘˜ Μ‚) = –4 Γ— 4 + (–6) Γ— 6 + (–8) Γ— 8 = –16 – 36 – 64 = – 116 So, Shortest distance = |(((𝑏_1 ) βƒ— Γ— (𝑏_2 ) βƒ— ).((π‘Ž_2 ) βƒ— βˆ’ (π‘Ž_1 ) βƒ— ))/|(𝑏_1 ) βƒ— Γ— (𝑏_2 ) βƒ— | | = |( βˆ’116)/√116| = βˆšπŸπŸπŸ” Therefore, shortest distance between the given two lines is (3√2)/2.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.