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Supplementary Example 1 If 𝑎 ⃗ = 2𝑖 ̂ − 3𝑗 ̂ + 4𝑘 ̂ , 𝑏 ⃗ = 𝑖 ̂ + 2𝑗 ̂ – 3𝑘 ̂ , 𝑐 ⃗ = 3𝑖 ̂ + 4𝑗 ̂ – 𝑘 ̂, then find 𝑎 ⃗.(𝑏 ⃗ × 𝑐 ⃗) & (𝑎 ⃗ × 𝑏 ⃗).𝑐 ⃗ . Is 𝑎 ⃗.(𝑏 ⃗ × 𝑐 ⃗) = (𝑎 ⃗ × 𝑏 ⃗).𝑐 ⃗ ? Given, 𝑎 ⃗ = 2𝑖 ̂ − 3𝑗 ̂ + 4𝑘 ̂ , 𝑏 ⃗ = 𝑖 ̂ + 2𝑗 ̂ – 3𝑘 ̂ , 𝑐 ⃗ = 3𝑖 ̂ + 4𝑗 ̂ – 𝑘 ̂ 𝑎 ⃗.(𝑏 ⃗ × 𝑐 ⃗) = [𝑎 ⃗" " 𝑏 ⃗" " 𝑐 ⃗ ] = |■8(2&−3&[email protected]&2&−[email protected]&4&−1)| = 2[(2×−1)−(4×−3) ] − (−3) [(1×−1)−(3×−3) ] + 4[(1×4)−(3×2)] = 2 [−2+12]+3(−1+9)+4[4−6] = 2(10) + 3 (8) + 4(–2) = 20 + 24 – 8 = 36 Finding (𝑎 ⃗ × 𝑏 ⃗).𝑐 ⃗ = 𝑐 ⃗.(𝑎 ⃗ × 𝑏 ⃗) = [𝑐 ⃗" " 𝑎 ⃗" " 𝑏 ⃗ ] = |■8(3&4&−[email protected]&−3&[email protected]&2&−3)| = 3[(−3×−3)−(2×4) ] − 4[(2×−3)−(1×4) ] + (–1)[(2×2)−(1×−3)] = 3[9−8]−4(−6−4)−[4+3] = 3(1) – 4(–10) – 7 = 3 + 40 – 7 = 36 Hence , 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.