Ex 10.4, 7 - Show that a x (b + c) =  a x b + a x c - Ex 10.4

Ex 10.4, 7 - Chapter 10 Class 12 Vector Algebra - Part 2
Ex 10.4, 7 - Chapter 10 Class 12 Vector Algebra - Part 3

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Transcript

Ex 10.4, 7 Let the vectors 𝑎 ⃗ 𝑏 ⃗, 𝑐 ⃗ be given as 𝑎_1 𝑖 ̂ + 𝑎_2 𝑗 ̂ +𝑎_3 𝑘 ̂, 𝑏_1 𝑖 ̂ + 𝑏_2 𝑗 ̂ +𝑏_3 𝑘 ̂, 𝑐_1 𝑖 ̂ + 𝑐_2 𝑗 ̂ +𝑐_3 𝑘 ̂ Then show that 𝑎 ⃗ × (𝑏 ⃗ + 𝑐 ⃗) =𝑎 ⃗ ×𝑏 ⃗ + 𝑎 ⃗ × 𝑐 ⃗. Let 𝑎 ⃗ = 𝑎_1 𝑖 ̂ + 𝑎_2 𝑗 ̂ +𝑎_3 𝑘 ̂ 𝑏 ⃗ = 𝑏_1 𝑖 ̂ + 𝑏_2 𝑗 ̂ +𝑏_3 𝑘 ̂, 𝑐 ⃗ = 𝑐_1 𝑖 ̂ + 𝑐_2 𝑗 ̂ + 𝑐_3 𝑘 ̂ We need to show : 𝑎 ⃗ × (𝑏 ⃗ + 𝑐 ⃗) = 𝑎 ⃗ × 𝑏 ⃗ + 𝑎 ⃗ × 𝑐 ⃗ RHS (𝑎 ⃗ × 𝑏 ⃗) = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@𝑎1&𝑎2&𝑎3@𝑏1&𝑏2&𝑏3)| = (𝑎2 𝑏3 − 𝑏2 𝑎3) 𝑖 ̂ − (𝑎1 𝑏3 − 𝑏1 𝑎3) 𝑗 ̂ + (𝑎1 𝑏2 − 𝑏1 𝑎2) 𝑘 ̂ (𝑎 ⃗ × 𝑐 ⃗) = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@𝑎1&𝑎2&𝑎3@𝑐1&𝑐2&𝑐3)| = (𝑎2" " 𝑐3 − 𝑐2" " 𝑎3) 𝑖 ̂ − (𝑎1" " 𝑐3 − 𝑐1" " 𝑎3) 𝑗 ̂ + (𝑎1" " 𝑐2 − 𝑐1" " 𝑎2) 𝑘 ̂ (𝒂 ⃗ × 𝒃 ⃗) + (𝒂 ⃗ × 𝒄 ⃗) = [𝑎2" " 𝑏3−𝑏2 𝑎3+𝑎2 𝑐3−𝑐2𝑎3] 𝑖 ̂ − [𝑎1" " 𝑏3−𝑏1 𝑎3+𝑎1 𝑐3−𝑐1𝑎3] 𝑗 ̂ + [𝑎1" " 𝑏2−𝑏1 𝑎2+𝑎1 𝑐2−𝑐1𝑎2] 𝑘 ̂ Since the corresponding components are equal, So, 𝑎 ⃗ × (𝑏 ⃗ + 𝑐 ⃗) = 𝑎 ⃗ × 𝑏 ⃗ + 𝑎 ⃗ × 𝑐 ⃗ Hence proved. Ex 10.4, 7 Let the vectors , be given as 1 + 2 + 3 , 1 + 2 + 3 , 1 + 2 + 3 Then show that ( + ) = + . Let = 1 + 2 + 3 = 1 + 2 + 3 , = 1 + 2 + 3 We need to show : ( + ) = + LHS ( + ) = ( 1 + 1) + ( 2 + 2) + ( 3 + 3) ( + ) = 1 ( 1+ 1) 2 ( 2+ 2) 3 ( 3+ 3) = 2 ( 3 + 3) ( 2 + 2) 3 1 ( 3 + 3) ( 1 + 1) 3 + 1 ( 2 + 2) ( 1 + 1) 2 = 2 3+ 2 3 2 3 2 3 1 3 1 3+ 1 3 1 3 + 1 2+ 1 2 1 2 1 2

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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.