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Example 12 - Class 12 Chapter 11 - Find distance between lines

Example 12 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 12 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3 Example 12 - Chapter 11 Class 12 Three Dimensional Geometry - Part 4


Transcript

Example 12 Find the distance between the lines 𝑙_1 and 𝑙_2 given by π‘Ÿ βƒ— = 𝑖 Μ‚ + 2𝑗 Μ‚ – 4π‘˜ Μ‚ + πœ† (2π’Š Μ‚ + 3𝒋 Μ‚ + 6π’Œ Μ‚ ) and π‘Ÿ βƒ— = 3𝑖 Μ‚ + 3𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚ + ΞΌ (2π’Š Μ‚ + 3𝒋 Μ‚ + 6π’Œ Μ‚)Since they are same, they are parallel lines Distance between two parallel lines with vector equations π‘Ÿ βƒ— = (π‘Ž_1 ) βƒ— + πœ†π‘ βƒ— and π‘Ÿ βƒ— = (π‘Ž_2 ) βƒ— + μ𝑏 βƒ— is |(𝑏 βƒ— Γ— ((π‘Ž_2 ) βƒ— βˆ’ (π‘Ž_1 ) βƒ—))/|𝑏 βƒ— | | π‘Ÿ βƒ— = (𝑖 Μ‚ + 2𝑗 Μ‚ βˆ’ 4π‘˜ Μ‚) + πœ† (2π’Š Μ‚ + 3𝒋 Μ‚ + 6π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž1) βƒ— + πœ† 𝑏 βƒ—, (π‘Ž1) βƒ— = 1𝑖 Μ‚ + 2𝑗 Μ‚ – 4π‘˜ Μ‚ & 𝑏 βƒ— = 2𝑖 Μ‚ + 3𝑗 Μ‚ + 6π‘˜ Μ‚ π‘Ÿ βƒ— = (3𝑖 Μ‚ + 3𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚) + πœ‡ (2π’Š Μ‚ + 3𝒋 Μ‚ + 6π’Œ Μ‚) Comparing with π‘Ÿ βƒ— = (π‘Ž2) βƒ— + πœ‡π‘ βƒ—, (π‘Ž2) βƒ— = 3𝑖 Μ‚ + 3𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚ & 𝑏 βƒ— = 2𝑖 Μ‚ + 3𝑗 Μ‚ + 6π‘˜ Μ‚ Now, ((π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ—) = (3𝑖 Μ‚ + 3𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 2𝑗 Μ‚ βˆ’ 4π‘˜ Μ‚) = (3 βˆ’ 1) 𝑖 Μ‚ + (3 βˆ’ 2)𝑗 Μ‚ + ( βˆ’ 5 + 4)π‘˜ Μ‚ = 2π’Š Μ‚ + 1𝒋 Μ‚ βˆ’ 1π’Œ Μ‚ Magnitude of 𝑏 βƒ— = √(22 + 32 + 62) |𝒃 βƒ— | = √(4+9+36) = √49 = 7 Also, 𝒃 βƒ— Γ— ((π’‚πŸ) βƒ— βˆ’ (π’‚πŸ) βƒ—) = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@2&3&[email protected]&1&βˆ’1)| = 𝑖 Μ‚ [(3Γ—βˆ’1)βˆ’(1Γ—6)] βˆ’ 𝑗 Μ‚ [(2Γ—βˆ’1)βˆ’(2Γ—6)] + π‘˜ Μ‚ [(2Γ—1)βˆ’(2Γ—3)] = 𝑖 Μ‚ [βˆ’3βˆ’6] βˆ’ 𝑗 Μ‚ [βˆ’2βˆ’12] + π‘˜ Μ‚ [2βˆ’6] = 𝑖 Μ‚ (–9) βˆ’ 𝑗 Μ‚ (–14) + π‘˜ Μ‚(βˆ’4) = βˆ’πŸ—π’Š Μ‚ + 14𝒋 Μ‚ βˆ’ 4π’Œ Μ‚ Now, |𝒃 βƒ—" Γ— (" (π’‚πŸ) βƒ—" βˆ’ " (π’‚πŸ) βƒ—")" | = √((βˆ’9)^2+(14)^2+(βˆ’4)^2 ) = √(81+196+16) = βˆšπŸπŸ—πŸ‘ So, distance = |(𝑏 βƒ— Γ— ((π‘Ž_2 ) βƒ— βˆ’ (π‘Ž_1 ) βƒ—))/|𝑏 βƒ— | | = |√293/7| = βˆšπŸπŸ—πŸ‘/πŸ• Therefore, the distance between the given two parallel lines is √293/7.

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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.