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Example 30 (Supplementary NCERT) - Prove [a+b b+c c+a] = 2[a b c]

Example 30 (Supplementary NCERT) - Chapter 10 Class 12 Vector Algebra - Part 2

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Example 30 (Supplementary NCERT) Prove that [β– 8(π‘Ž βƒ—" + " 𝑏 βƒ—&𝑏 βƒ—+𝑐 βƒ—&𝑐 βƒ—+π‘Ž βƒ— )] = 2 [π‘Ž βƒ— , 𝑏 βƒ—, 𝑐 βƒ—] Solving LHS [β– 8(π‘Ž βƒ—" + " 𝑏 βƒ—&𝑏 βƒ—+𝑐 βƒ—&𝑐 βƒ—+π‘Ž βƒ— )] = (π‘Ž βƒ—" + " 𝑏 βƒ—). ["(" 𝑏 βƒ—+𝑐 βƒ—") " Γ— " (" 𝑐 βƒ—+π‘Ž βƒ—)] = (π‘Ž βƒ—" + " 𝑏 βƒ—). ["(" 𝑏 ⃗×𝑐 βƒ—") + (" 𝑏 βƒ—Γ—π‘Ž βƒ—)+"(" 𝒄 ⃗×𝒄 βƒ—") + (" 𝑐 βƒ—Γ—π‘Ž βƒ—)] = "(" π‘Ž βƒ—+𝑏 βƒ—")." [(𝑏 ⃗×𝑐 βƒ—") + (" 𝑏 βƒ—Γ—π‘Ž βƒ— ) "+ 0 + (" 𝑐 βƒ—" Γ— " π‘Ž βƒ—") " ] = π‘Ž βƒ—. (𝑏 βƒ— Γ— 𝑐 βƒ—) + 𝑏 βƒ—.(𝑏 βƒ— Γ— 𝑐 βƒ—) + π‘Ž βƒ—. (𝑏 βƒ— Γ— π‘Ž βƒ—) + 𝑏 βƒ—.(𝑏 βƒ— Γ— π‘Ž βƒ—) + π‘Ž βƒ—.(𝑐 βƒ— Γ— π‘Ž βƒ—) + 𝑏 βƒ—.(𝑐 βƒ— Γ— π‘Ž βƒ—) = [π‘Ž βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] + [𝒃 βƒ—", " 𝒃 βƒ—", " 𝒄 βƒ— ] + [𝒂 βƒ—", " 𝒃 βƒ—", " 𝒂 βƒ— ] + [𝒃 βƒ—", " 𝒃 βƒ—", " 𝒂 βƒ— ] + [𝒂 βƒ—", " 𝒄 βƒ—", " 𝒂 βƒ— ] + [𝑏 βƒ—", " 𝑐 βƒ—", " π‘Ž βƒ— ] 𝑐 βƒ— Γ— 𝑐 βƒ— = |𝑐 βƒ— ||𝑐 βƒ— | sin 0 𝑛 Μ‚ = 0 [𝑏 βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] = [𝑐 βƒ—", " 𝑏 βƒ—", " 𝑏 βƒ— ] = 𝑐 βƒ—. (𝑏 βƒ— Γ— 𝑏 βƒ—) As (𝑏 βƒ— Γ— 𝑏 βƒ—) = 0 βƒ— = 𝑐 βƒ— . 0 βƒ— = 0 Using Prop: [𝑏 βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] = 0 [π‘Ž βƒ—", " 𝑏 βƒ—", " π‘Ž βƒ— ] = 0 [𝑏 βƒ—", " 𝑏 βƒ—", " π‘Ž βƒ— ] = 0 [π‘Ž βƒ—", " 𝑐 βƒ—", " π‘Ž βƒ— ] = 0 = [π‘Ž βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] + 0 + 0 + 0 + 0 + [𝑏 βƒ—", " 𝑐 βƒ—", " π‘Ž βƒ— ] = [π‘Ž βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] + [𝒃 βƒ—", " 𝒄 βƒ—", " 𝒂 βƒ— ] = [π‘Ž βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] + [𝒂 βƒ—", " 𝒃 βƒ—", " 𝒄 βƒ— ] = 2[π‘Ž βƒ—", " 𝑏 βƒ—", " 𝑐 βƒ— ] = R.H.S 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.