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Get live Maths 1-on-1 Classs - Class 6 to 12


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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&[email protected]&2&−[email protected]&−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&[email protected]−2&3&−[email protected]&−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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.