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Supplementary Example 6 For any three vectors ∴ 𝑎 ⃗, 𝑏 ⃗, and 𝑐 ⃗, prove that [■8(𝑎 ⃗" + " 𝑏 ⃗&𝑏 ⃗+𝑐 ⃗&𝑐 ⃗+𝑎 ⃗ )] = 2 [𝑎 ⃗ , 𝑏 ⃗, 𝑐 ⃗] To prove [■8(𝑎 ⃗" + " 𝑏 ⃗&𝑏 ⃗+𝑐 ⃗&𝑐 ⃗+𝑎 ⃗ )] = 2 [𝑎 ⃗ , 𝑏 ⃗, 𝑐 ⃗] Solving LHS [■8(𝑎 ⃗" + " 𝑏 ⃗&𝑏 ⃗+𝑐 ⃗&𝑐 ⃗+𝑎 ⃗ )] = (𝑎 ⃗" + " 𝑏 ⃗). ["(" 𝑏 ⃗+𝑐 ⃗") " × " (" 𝑐 ⃗+𝑎 ⃗)] = (𝑎 ⃗" + " 𝑏 ⃗). ["(" 𝑏 ⃗×𝑐 ⃗") + (" 𝑏 ⃗×𝑎 ⃗)+"(" 𝒄 ⃗×𝒄 ⃗") + (" 𝑐 ⃗×𝑎 ⃗)] = "(" 𝑎 ⃗+𝑏 ⃗")." [(𝑏 ⃗×𝑐 ⃗") + (" 𝑏 ⃗×𝑎 ⃗ ) "+ 0 + (" 𝑐 ⃗" × " 𝑎 ⃗") " ] = 𝑎 ⃗. (𝑏 ⃗ × 𝑐 ⃗) + 𝑏 ⃗.(𝑏 ⃗ × 𝑐 ⃗) + 𝑎 ⃗. (𝑏 ⃗ × 𝑎 ⃗) + 𝑏 ⃗.(𝑏 ⃗ × 𝑎 ⃗) + 𝑎 ⃗.(𝑐 ⃗ × 𝑎 ⃗) + 𝑏 ⃗.(𝑐 ⃗ × 𝑎 ⃗) = [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + [𝒃 ⃗", " 𝒃 ⃗", " 𝒄 ⃗ ] + [𝒂 ⃗", " 𝒃 ⃗", " 𝒂 ⃗ ] + [𝒃 ⃗", " 𝒃 ⃗", " 𝒂 ⃗ ] + [𝒂 ⃗", " 𝒄 ⃗", " 𝒂 ⃗ ] + [𝑏 ⃗", " 𝑐 ⃗", " 𝑎 ⃗ ] [𝑏 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] = [𝑐 ⃗", " 𝑏 ⃗", " 𝑏 ⃗ ] = 𝑐 ⃗. (𝑏 ⃗ × 𝑏 ⃗) As (𝑏 ⃗ × 𝑏 ⃗) = 0 ⃗ = 𝑐 ⃗ . 0 ⃗ = 0 ⃗ Using Prop: [𝑏 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] = 0 [𝑎 ⃗", " 𝑏 ⃗", " 𝑎 ⃗ ] = 0 [𝑏 ⃗", " 𝑏 ⃗", " 𝑎 ⃗ ] = 0 [𝑎 ⃗", " 𝑐 ⃗", " 𝑎 ⃗ ] = 0 = [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + 0 + 0 + 0 + 0 + [𝑏 ⃗", " 𝑐 ⃗", " 𝑎 ⃗ ] = [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + [𝒃 ⃗", " 𝒄 ⃗", " 𝒂 ⃗ ] = [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + [𝒂 ⃗", " 𝒃 ⃗", " 𝒄 ⃗ ]

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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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.