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If a = i Μ‚ - j Μ‚ + 7k Μ‚ π‘Žπ‘›π‘‘ b βƒ—= 5i Μ‚ - j Μ‚ + Ξ»k Μ‚, then find the value of πœ† so that the vectors a βƒ— + b βƒ— andΒ  a βƒ— - b βƒ— are orthogonal

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Transcript

Question 23 (Choice 1) If π‘Ž βƒ—=𝑖 Μ‚βˆ’π‘— Μ‚+7π‘˜ Μ‚ π‘Žπ‘›π‘‘ 𝑏 βƒ—=5𝑖 Μ‚βˆ’π‘— Μ‚+"πœ†" π‘˜ Μ‚, then find the value of πœ† so that the vectors π‘Ž βƒ—+𝑏 βƒ— and π‘Ž βƒ—βˆ’π‘ βƒ— are orthogonal Two vectors 𝑝 βƒ— and π‘ž βƒ— are orthogonal (perpendicular) if their scalar product is zero, i.e. 𝒑 βƒ— . 𝒒 βƒ— = 0 Finding (𝒂 βƒ— + 𝒃 βƒ—) and (𝒂 βƒ— βˆ’ 𝒃 βƒ—) (𝒂 βƒ— + 𝒃 βƒ—) = (1 + 5) 𝑖 Μ‚ + (βˆ’1 + (βˆ’1)) 𝑗 Μ‚ + (7 + πœ†) π‘˜ Μ‚ = 6π’Š Μ‚ βˆ’ 2𝒋 Μ‚ + (7 + πœ†)π’Œ Μ‚ (𝒂 βƒ— βˆ’ 𝒃 βƒ—) = (1 βˆ’ 5) 𝑖 Μ‚ + (βˆ’1 βˆ’ (βˆ’1)) 𝑗 Μ‚ + (7 βˆ’ πœ†) π‘˜ Μ‚ = βˆ’4π’Š Μ‚ + 0𝒋 Μ‚ + (7 βˆ’ πœ†)π’Œ Μ‚ Since (π‘Ž βƒ— + 𝑏 βƒ—) and (π‘Ž βƒ— βˆ’ 𝑏 βƒ—) are perpendicular to each other. Thus, (𝒂 βƒ— + 𝒃 βƒ—) . (𝒂 βƒ— βˆ’ 𝒃 βƒ—) = 0 (6𝑖 Μ‚ βˆ’ 2𝑗 Μ‚ + (7 + πœ†)π‘˜ Μ‚) . (βˆ’4𝑖 Μ‚ + 0𝑗 Μ‚ + (7 βˆ’ πœ†)π‘˜ Μ‚) = 0 (6 Γ— βˆ’4) + (βˆ’2 Γ— 0) + (7 + πœ†) (7 βˆ’ πœ†) = 0 βˆ’24 + 0 + 72 βˆ’ πœ†2 = 0 βˆ’24 + 0 + 49 βˆ’ πœ†2 = 0 25 = πœ†2 πœ†2 = 25 πœ† = Β±5

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