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Example 15 - Show vectors a + b and a - b are perpendicular

Example 15 - Chapter 10 Class 12 Vector Algebra - Part 2

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Example 15 If π‘Ž βƒ— = 5𝑖 Μ‚ βˆ’ 𝑗 Μ‚ βˆ’ 3π‘˜ Μ‚ and 𝑏 βƒ— = 𝑖 Μ‚ + 3𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚ , then show that the vectors π‘Ž βƒ— + 𝑏 βƒ— and π‘Ž βƒ— βˆ’ 𝑏 βƒ— are perpendicular. Two vectors 𝑝 βƒ— and π‘ž βƒ— are 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—) = (5 + 1) 𝑖 Μ‚ + (βˆ’1 + 3) 𝑗 Μ‚ + (βˆ’3 + (βˆ’5)) π‘˜ Μ‚ = 6π’Š Μ‚ + 2𝒋 Μ‚ βˆ’ 8π’Œ Μ‚ (𝒂 βƒ— βˆ’ 𝒃 βƒ—) = (5 βˆ’ 1) 𝑖 Μ‚ + (βˆ’1 βˆ’ 3) 𝑗 Μ‚ + (βˆ’3 βˆ’ (βˆ’5)) π‘˜ Μ‚ = 4π’Š Μ‚ βˆ’ 4𝒋 Μ‚ + 2π’Œ Μ‚ We have to show that (π‘Ž βƒ— + 𝑏 βƒ—) and (π‘Ž βƒ— βˆ’ 𝑏 βƒ—) are perpendicular to each other. So, we need to show (𝒂 βƒ— + 𝒃 βƒ—) . (𝒂 βƒ— βˆ’ 𝒃 βƒ—) = 0 Solving LHS (𝒂 βƒ— + 𝒃 βƒ—) . (𝒂 βƒ— βˆ’ 𝒃 βƒ—) = (6𝑖 Μ‚ + 2𝑗 Μ‚ βˆ’ 8π‘˜ Μ‚) . (4𝑖 Μ‚ βˆ’ 4𝑗 Μ‚ + 2π‘˜ Μ‚) = (6 Γ— 4) + (2 Γ— βˆ’4) + (βˆ’8 Γ— 2) = 24 βˆ’ 8 βˆ’16 = 0 Since (π‘Ž βƒ— + 𝑏 βƒ—) . (π‘Ž βƒ— βˆ’ 𝑏 βƒ—) = 0 Hence, (π‘Ž βƒ— + 𝑏 βƒ—) is perpendicular to (π‘Ž βƒ— βˆ’ 𝑏 βƒ—)

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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.