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Supplementary Example 4 Show that the vectors 𝑎 ⃗ = 𝑖 ̂ − 2𝑗 ̂ + 3𝑘 ̂, 𝑏 ⃗ = −2𝑖 ̂ + 3𝑗 ̂ − 4𝑘 ̂ and 𝑐 ⃗ = 𝑖 ̂ − 3𝑗 ̂ + 𝜆𝑘 ̂ are co-planar if 𝜆 = 5 Three vectors 𝑎 ⃗, 𝑏 ⃗, 𝑐 ⃗ are coplanar if [𝒂 ⃗" " 𝒃 ⃗" " 𝒄 ⃗ ] = 0 Given, 𝑎 ⃗ = 𝑖 ̂ − 2𝑗 ̂ + 3𝑘 ̂ 𝑏 ⃗ = −2𝑖 ̂ + 3𝑗 ̂ − 4𝑘 ̂ 𝑐 ⃗ = 𝑖 ̂ − 3𝑗 ̂ + λ𝑘 ̂ [𝑎 ⃗" " 𝑏 ⃗" " 𝑐 ⃗ ] = |■8(1&−2&[email protected]−2&3&−[email protected]&−3&𝜆)| 0 = 1[(3×λ)−(−3×−4) ] − (−2) [(−2×λ)−(1×−4) ] + 3[(2×−3)−(1×3) ] 0 = 1 [3λ−12]+2[−2λ−(−4)]+3[6−3] 0 = 3λ – 12 – 4λ + 8 + 9 0 = – λ + 5 𝝀 = 5 Therefore, 𝑎 ⃗,𝑏,𝑐 ⃗ are coplanar if λ = 5

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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.