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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise

Transcript

Ex 11.2, 14 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, ((𝑏1) βƒ— Γ— (𝑏2) βƒ—) . ((π‘Ž2) βƒ— – (π‘Ž1) 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’9)/(3√2)| = 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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.