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Misc 15 - Prove that vectors (a + b) . (a + b) = |a|^2 + |b|^2

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

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Misc 15 Prove that (𝑎 ⃗ + 𝑏 ⃗) ⋅ (𝑎 ⃗ + 𝑏 ⃗) =|𝑎 ⃗|2 + |𝑏 ⃗|2 , if and only if 𝑎 ⃗, 𝑏 ⃗ are perpendicular, given 𝑎 ⃗ ≠ 0 ⃗, 𝑏 ⃗ ≠ 0 ⃗ (𝑎 ⃗ + 𝑏 ⃗) ⋅ (𝑎 ⃗ + 𝑏 ⃗) = 𝑎 ⃗ . 𝑎 ⃗ + 𝑎 ⃗ . 𝑏 ⃗ + 𝒃 ⃗ . 𝒂 ⃗ + 𝑏 ⃗ . 𝑏 ⃗ = 𝑎 ⃗ . 𝑎 ⃗ + 𝑎 ⃗ . 𝑏 ⃗ + 𝒂 ⃗ . 𝒃 ⃗ + 𝑏 ⃗ . 𝑏 ⃗ = 𝒂 ⃗ . 𝒂 ⃗ + 2𝑎 ⃗ . 𝑏 ⃗ + 𝒃 ⃗ . 𝒃 ⃗ =|𝒂 ⃗|2 + 2𝑎 ⃗ . 𝑏 ⃗ + |𝒃 ⃗|2 Since 𝑎 ⃗ and 𝑏 ⃗ are perpendicular, 𝒂 ⃗ . 𝒃 ⃗ = 0 (Using prop: 𝑎 ⃗.𝑏 ⃗ = 𝑏 ⃗.𝑎 ⃗) (Using prop: 𝑎 ⃗.𝑎 ⃗ =|𝑎 ⃗ |^2) Putting 𝑎 ⃗ . 𝑏 ⃗ = 0 in (1) (𝒂 ⃗ + 𝒃 ⃗) . (𝒂 ⃗ + 𝒃 ⃗) = |𝑎 ⃗|2 + 2.(0) + |𝑏 ⃗|2 = |𝒂 ⃗|2 + |𝒃 ⃗|2 Hence proved

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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.