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Example 11 - Chapter 11 Class 12 - Find shortest distance

Example 11 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 11 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Example 11 Find the shortest distance between the lines l1 and l2 whose vector equations are π‘Ÿ βƒ— = 𝑖 Μ‚ + 𝑗 Μ‚ + πœ†(2𝑖 Μ‚ βˆ’ 𝑗 Μ‚ + π‘˜ Μ‚ ) and π‘Ÿ βƒ— = 2𝑖 Μ‚ + 𝑗 Μ‚ – π‘˜ Μ‚ + πœ‡ (3𝑖 Μ‚ – 5𝑗 Μ‚ + 2π‘˜ Μ‚ )Shortest distance between 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 |(((𝑏1) βƒ— Γ— (𝑏2) βƒ— ).((π‘Ž2) βƒ— βˆ’ (π‘Ž1) βƒ— ))/|(𝑏1) βƒ— Γ— (𝑏2) βƒ— | | 𝒓 βƒ— = (π’Š Μ‚ + 𝒋 Μ‚) + πœ† (2π’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‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚ & (𝑏1) βƒ— = 2𝑖 Μ‚ – 1𝑗 Μ‚ + 1π‘˜ Μ‚ 𝒓 βƒ— = (2π’Š Μ‚ + 𝒋 Μ‚ βˆ’ π’Œ Μ‚) + 𝝁 (3π’Š Μ‚ βˆ’ 5𝒋 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— = 3𝑖 Μ‚ βˆ’ 5𝑗 Μ‚ + 2π‘˜ Μ‚ Now (π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ— = (2𝑖 Μ‚ + 1𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚) = (2 βˆ’ 1) 𝑖 Μ‚ + (1 βˆ’ 1)𝑗 Μ‚ + (βˆ’1 βˆ’ 0) π‘˜ Μ‚ = 1π’Š Μ‚ + 0𝒋 Μ‚ βˆ’ 1π’Œ Μ‚ (π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@2& βˆ’1&1@3& βˆ’5&2)| = 𝑖 Μ‚ [(βˆ’1Γ—2)βˆ’(βˆ’5Γ—1)] βˆ’ 𝑗 Μ‚ [(2Γ—2)βˆ’(3Γ—1)] + π‘˜ Μ‚[(2Γ—βˆ’5)βˆ’(3Γ—βˆ’1)] = 𝑖 Μ‚ [βˆ’2+5] βˆ’ 𝑗 Μ‚ [4βˆ’3] + π‘˜ Μ‚ [βˆ’10+3] = 𝑖 Μ‚ (3) βˆ’ 𝑗 Μ‚ (1) + π‘˜ Μ‚(βˆ’7) = 3π’Š Μ‚ βˆ’ 𝒋 Μ‚ βˆ’ 7π’Œ Μ‚ Magnitude of ((𝑏1) βƒ— Γ— (𝑏2) βƒ—) = √(32+(βˆ’1)2+(βˆ’7)^2 ) |(π’ƒπŸ) βƒ—Γ— (π’ƒπŸ) βƒ— | = √(9+1+49) = βˆšπŸ“πŸ— Also, ((π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ—) .((π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ—) = (3𝑖 Μ‚ βˆ’ 𝑗 Μ‚ βˆ’ 7π‘˜ Μ‚) . (1𝑖 Μ‚ + 0𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) = (3 Γ— 1) + (βˆ’1 Γ— 0) + (βˆ’7 Γ— βˆ’1) = 3 + 0 + 7 = 10 ∴ Shortest distance = |(((𝑏1) βƒ— Γ— (𝑏2) βƒ— ).((π‘Ž2) βƒ— βˆ’ (π‘Ž1) βƒ— ))/|(𝑏1) βƒ— Γ— (𝑏2) βƒ— | | = |10/√59| = 𝟏𝟎/βˆšπŸ“πŸ— .

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.