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Misc 13 - Scalar product of vector i + j + k with unit vector

Misc 13 - Chapter 10 Class 12 Vector Algebra - Part 2
Misc 13 - Chapter 10 Class 12 Vector Algebra - Part 3

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Misc 13 The scalar product of the vector 𝑖 Μ‚ + 𝑗 Μ‚ + π‘˜ Μ‚ with a unit vector along the sum of vectors 2𝑖 Μ‚ + 4𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚ and λ𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚ is equal to one. Find the value of Ξ». Let 𝒂 βƒ— = 𝑖 Μ‚ + 𝑗 Μ‚ + π‘˜ Μ‚ 𝒃 βƒ— = 2𝑖 Μ‚ + 4𝑗 Μ‚ – 5π‘˜ Μ‚ 𝒄 βƒ— = πœ† 𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚ (𝒃 βƒ— + 𝒄 βƒ—) = (2 + πœ†) 𝑖 Μ‚ + (4 + 2) 𝑗 Μ‚ + (βˆ’5 + 3) π‘˜ Μ‚ = (2 + πœ†) π’Š Μ‚ + 6𝒋 Μ‚ βˆ’ 2π’Œ Μ‚ Let 𝒓 Μ‚ be unit vector along (𝑏 βƒ— + 𝑐 βƒ—) π‘Ÿ Μ‚ = 1/(π‘€π‘Žπ‘”π‘›π‘–π‘‘π‘’π‘‘π‘’ π‘œπ‘“ (𝑏 βƒ—" + " 𝑐 βƒ—)) Γ— (𝑏 βƒ— + 𝑐 βƒ—) π‘Ÿ Μ‚ = 1/√((2 + πœ†)^2 + 6^2 + (βˆ’2)^2 ) Γ— ((2 + πœ†) 𝑖 Μ‚ + 6𝑗 Μ‚ βˆ’ 2π‘˜ Μ‚) π‘Ÿ Μ‚ = 1/√(2^2 + πœ†^2 + 4πœ† + 36 + 4) Γ— ((2 + πœ†) 𝑖 Μ‚ + 6𝑗 Μ‚ βˆ’ 2π‘˜ Μ‚) 𝒓 Μ‚ = 𝟏/√(𝝀^𝟐 + πŸ’π€ +πŸ’πŸ’) Γ— ((2 + πœ†) π’Š Μ‚ + 6𝒋 Μ‚ βˆ’ 2π’Œ Μ‚) Given, 𝒂 βƒ—. (𝒓 Μ‚) = 1 (1𝑖 Μ‚ + 1𝑗 Μ‚ + 1π‘˜ Μ‚). (1/√(πœ†^2 + 4πœ† +44) " Γ— ((2 + πœ†) " 𝑖 Μ‚" + 6" 𝑗 Μ‚" βˆ’ 2" π‘˜ Μ‚")" ) = 1 1/√(πœ†^2 + 4πœ† +44) (1𝑖 Μ‚ + 1𝑗 Μ‚ + 1π‘˜ Μ‚).((πœ† +2) 𝑖 Μ‚ + 6𝑗 Μ‚ βˆ’ 2π‘˜ Μ‚) = 1 (1𝑖 Μ‚ + 1𝑗 Μ‚ + 1π‘˜ Μ‚).((πœ† +2) 𝑖 Μ‚ + 6𝑗 Μ‚ βˆ’ 2π‘˜ Μ‚) = √(πœ†^2 + 4πœ† +44) 1.(πœ† + 2) + 1.6 + 1.(βˆ’2) = √(πœ†^2 + 4πœ† +44) πœ† + 2 + 6 βˆ’ 2 = √(πœ†^2 + 4πœ† +44) πœ† + 6 = √(𝝀^𝟐 + πŸ’π€ +πŸ’πŸ’) Squaring both sides (πœ† + 6)2 = (√(πœ†^2 + 4πœ† +44))^2 πœ†2 + 36 + 12πœ† = πœ†^2 + 4πœ† +44 8πœ† = 8 πœ† = 8/8 πœ† = 1 So, πœ† = 1

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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.