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Ex 11.2, 12 Find the shortest distance between the lines π‘Ÿ βƒ— = (𝑖 Μ‚ + 2𝑗 Μ‚ + π‘˜ Μ‚) + πœ† (𝑖 Μ‚ βˆ’ 𝑗 Μ‚ + π‘˜ Μ‚) and π‘Ÿ βƒ— = (2𝑖 Μ‚ βˆ’ 𝑗 Μ‚ βˆ’ π‘˜ Μ‚) + πœ‡ (2𝑖 Μ‚ + 𝑗 Μ‚ + 2π‘˜ Μ‚) 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— | | Given, 𝒓 βƒ— = (π’Š Μ‚ + 2𝒋 Μ‚ + π’Œ Μ‚) + πœ†(π’Š Μ‚ βˆ’ 𝒋 Μ‚ + π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž1) βƒ— + πœ† (𝑏1) βƒ—, (π‘Ž1) βƒ— = 1𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚ & (𝑏1) βƒ— = 1𝑖 Μ‚ – 1𝑗 Μ‚ + 1π‘˜ Μ‚ 𝒓 βƒ— = (2π’Š Μ‚ βˆ’ 𝒋 Μ‚ βˆ’ π’Œ Μ‚) + 𝝁 (2π’Š Μ‚ + 𝒋 Μ‚ + 2π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž2) βƒ— + πœ‡(𝑏2) βƒ— , (π‘Ž2) βƒ— = 2𝑖 Μ‚ – 1𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚ & (𝑏2) βƒ— = 2𝑖 Μ‚ + 1𝑗 Μ‚ + 2π‘˜ Μ‚ Now, (π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ— = (2𝑖 Μ‚ βˆ’ 1𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚) = (2 βˆ’ 1) 𝑖 Μ‚ + (βˆ’1βˆ’ 2)𝑗 Μ‚ + (βˆ’1 βˆ’ 1) π‘˜ Μ‚ = 1π’Š Μ‚ βˆ’ 3𝒋 Μ‚ βˆ’ 2π’Œ Μ‚ (π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@1& βˆ’1&1@2&1&2)| = 𝑖 Μ‚ [(βˆ’1Γ— 2)βˆ’(1Γ—1)] βˆ’ 𝑗 Μ‚ [(1Γ—2)βˆ’(2Γ—1)] + π‘˜ Μ‚ [(1Γ—1)βˆ’(2Γ—βˆ’1)] = 𝑖 Μ‚ [βˆ’2βˆ’1] βˆ’ 𝑗 Μ‚ [2βˆ’2] + π‘˜ Μ‚ [1+2] = βˆ’3π’Š Μ‚ βˆ’ 0𝒋 Μ‚ + 3π’Œ Μ‚ Magnitude of ((𝑏1) βƒ— Γ— (𝑏2) βƒ—) = √((βˆ’3)2+(0)2+32) |(π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— | = √(9+0+9) = √18 = √(9 Γ— 2) = 3√𝟐 Also, ((π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ—) . ((π’‚πŸ) βƒ— – (π’‚πŸ) βƒ—) = (βˆ’ 3𝑖 Μ‚βˆ’0𝑗 Μ‚+3π‘˜ Μ‚).(1𝑖 Μ‚ βˆ’ 3𝑗 Μ‚ βˆ’ 2π‘˜ Μ‚) = (βˆ’3Γ—1)".(" 0Γ—βˆ’"3)" + (3 Γ— βˆ’2) = βˆ’3 βˆ’ 0 βˆ’ 6 = βˆ’9 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’πŸ—)/(πŸ‘βˆšπŸ)| = 3/√2 = 3/√2 Γ— √2/√2 = (πŸ‘βˆšπŸ)/𝟐 Therefore, shortest distance between the given two lines is (3√2)/2.

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Davneet Singh

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 and Computer Science at Teachoo