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Transcript

Supplementary Exercise Q4 Show that (i) the vectors 𝑎 ⃗ = 2𝑖 ̂ − 𝑗 ̂ + 𝑘 ̂, 𝑏 ⃗ = 𝑖 ̂ + 2𝑗 ̂ − 3𝑘 ̂, and 𝑐 ⃗ = 3𝑖 ̂ − 4𝑗 ̂ + 5𝑘 ̂ are coplanar. Three vectors 𝑎 ⃗, 𝑏 ⃗, 𝑐 ⃗ are coplanar if [ 𝒂 ⃗, 𝒃 ⃗, 𝒄 ⃗ ] = 0 Given, 𝑎 ⃗ = 2𝑖 ̂ − 𝑗 ̂ + 𝑘 ̂ 𝑏 ⃗ = 𝑖 ̂ + 2𝑗 ̂ − 3𝑘 ̂ 𝑐 ⃗ = 3𝑖 ̂ − 4𝑗 ̂ + 5𝑘 ̂ [𝑎 ⃗,𝑏,𝑐 ⃗ ] = |■8(2&−1&1@1&2&−3@3&−4&5)| = 2[(2×5)−(−4×−3)] − (−1) [(1×5)−(3×−3) ] + 1[(1×−4)−(3×2) ] = 2 [10−12]+[5+9] + [−4−6] = –4 + 14 – 10 = 0 ∴ [𝒂 ⃗,𝒃,𝒄 ⃗ ] = 0 Therefore, 𝑎 ⃗,𝑏 and 𝑐 ⃗ are coplanar. Supplementary Exercise Q4 Show that (ii) the vectors 𝑎 ⃗ = 𝑖 ̂ − 2𝑗 ̂ + 3𝑘 ̂, 𝑏 ⃗ = −2𝑖 ̂ + 3𝑗 ̂ − 4𝑘 ̂, and 𝑐 ⃗ = 𝑖 ̂ − 3𝑗 ̂ + 5𝑘 ̂ are coplanar Three vectors 𝑎 ⃗, 𝑏 ⃗, 𝑐 ⃗ are coplanar if [ 𝒂 ⃗, 𝒃 ⃗, 𝒄 ⃗ ] = 0 Given, 𝑎 ⃗ = 𝑖 ̂ − 2𝑗 ̂ + 3𝑘 ̂ 𝑏 ⃗ = −2𝑖 ̂ + 3𝑗 ̂ − 4𝑘 ̂ 𝑐 ⃗ = 𝑖 ̂ − 3𝑗 ̂ + 5𝑘 ̂ [𝑎 ⃗,𝑏,𝑐 ⃗ ] = |■8(1&−2&3@−2&3&−4@1&−3&5)| = 1[(3×5)−(−3×4) ] − (−2) [(−2×5)−(1×−4) ] + 3[(−2×−3)−(1×3) ] = 1 [15−12]+[2(−10−(−4)] + 3 [6−3] = 1(3) + 2 (−6) + 3 (3) = 3 − 12 + 9 = 12 − 12 = 0 ∴ [𝒂 ⃗,𝒃,𝒄 ⃗ ] = 0 Therefore, 𝑎 ⃗,𝑏 and 𝑐 ⃗ are coplanar.

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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.