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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


Transcript

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.