Misc 15 - Chapter 10 Class 12 Vector Algebra

Last updated at April 16, 2024 by

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

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

Go Ad-free

Transcript

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

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo