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

Transcript

Misc 9 Find the shortest distance between lines π‘Ÿ βƒ— = 6𝑖 Μ‚ + 2𝑗 Μ‚ + 2π‘˜ Μ‚ + πœ† (𝑖 Μ‚ – 2𝑗 Μ‚ + 2π‘˜ Μ‚) and π‘Ÿ βƒ— = –4𝑖 Μ‚ – π‘˜ Μ‚ + πœ‡ (3𝑖 Μ‚ – 2𝑗 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 |(((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— = (6π’Š Μ‚ + 2𝒋 Μ‚ + 2π’Œ Μ‚) + πœ† (π’Š Μ‚ βˆ’ 2𝒋 Μ‚ + 2π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž1) βƒ— + πœ†(𝑏1) βƒ— , (π‘Ž1) βƒ— = 6𝑖 Μ‚ + 2𝑗 Μ‚ + 2π‘˜ Μ‚ & (𝑏1) βƒ— = 1𝑖 Μ‚ βˆ’ 2𝑗 Μ‚ + 2π‘˜ Μ‚ 𝒓 βƒ— = (βˆ’4π’Š Μ‚ βˆ’ π’Œ Μ‚) + 𝝁 (3π’Š Μ‚ βˆ’ 2𝒋 Μ‚ βˆ’ 2π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž2) βƒ— + πœ‡(𝑏2) βƒ— , (π‘Ž2) βƒ— = βˆ’ 4𝑖 Μ‚ + 0𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚ & (𝑏2) βƒ— = 3𝑖 Μ‚ βˆ’ 2𝑗 Μ‚ βˆ’ 2π‘˜ Μ‚ Now, ((π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ—) = (βˆ’4𝑖 Μ‚ + 0𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) βˆ’ (6𝑖 Μ‚ + 2𝑗 Μ‚ + 2π‘˜ Μ‚) = (βˆ’4 βˆ’ 6) 𝑖 Μ‚ + (0 βˆ’ 2)𝑗 Μ‚ + (βˆ’1 βˆ’ 2) π‘˜ Μ‚ = βˆ’ 10π’Š Μ‚ βˆ’ 2𝒋 Μ‚ βˆ’ 3π’Œ Μ‚ ((π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ—) = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@1& βˆ’2&2@3&βˆ’2&βˆ’2)| = 𝑖 Μ‚ [(βˆ’2Γ—βˆ’2)βˆ’(βˆ’2Γ—2)] βˆ’ 𝑗 Μ‚ [(1Γ—βˆ’2)βˆ’(3Γ—2)] + π‘˜ Μ‚ [(1Γ—βˆ’2)βˆ’(3Γ—βˆ’2)] = 𝑖 Μ‚ [ 4+4] βˆ’ 𝑗 Μ‚ [βˆ’2βˆ’6] + π‘˜ Μ‚ [βˆ’2+6] = 𝑖 Μ‚ (8) βˆ’ 𝑗 Μ‚ (βˆ’8) + π‘˜ Μ‚(4) = 8π’Š Μ‚ + 8𝒋 Μ‚ + 4π’Œ Μ‚ Magnitude of (𝑏1) βƒ— Γ— (𝑏2) βƒ— = √(8^2+8^2+4^2 ) |(π’ƒπŸ) βƒ— Γ— (π’ƒπŸ) βƒ— | = √(64+64+16) = √144 = 𝟏𝟐 Also, ((π’ƒπŸ) βƒ—Γ—(π’ƒπŸ) βƒ— ) . ((π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ— ) = (8𝑖 Μ‚ + 8𝑗 Μ‚ + 4π‘˜ Μ‚).(βˆ’ 10𝑖 Μ‚ βˆ’ 2𝑗 Μ‚ βˆ’ 3π‘˜ Μ‚) = (8 Γ— βˆ’ 10) + (8 Γ— βˆ’ 2) + (4 Γ— βˆ’ 3) = βˆ’ 80 + (βˆ’16) + (-12) = βˆ’ 108 Shortest distance = |(((𝑏1) βƒ— Γ— (𝑏2) βƒ— ) . ((π‘Ž2) βƒ— βˆ’ (π‘Ž1) βƒ— ))/|(𝑏1) βƒ— Γ— (𝑏2) βƒ— | | = |( βˆ’108)/12| = |βˆ’9| = 9 Therefore, the shortest distance between the given two lines is 9.

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Davneet Singh's photo - Teacher, Computer Engineer, Marketer
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.