Supplementary Exercise Q5 - Find value of if vectors are co-planar - Supplementary examples and questions from CBSE

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Supplementary Exercise Q5 Find the value of 𝜆 if the following vectors are co-planar (i) 𝑎﷯ = 2 𝑖﷯ − 𝑗﷯ + 𝑘﷯, 𝑏﷯ = 𝑖﷯ + 2 𝑗﷯ − 3 𝑘﷯, and 𝑐﷯ = 3 𝑖﷯ − 𝜆 𝑗﷯ + 5 𝑘﷯ Three vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ are coplanar if 𝒂﷯ 𝒃﷯ 𝒄﷯﷯ = 0 Given, 𝑎﷯ = 2 𝑖﷯ − 𝑗﷯ + 𝑘﷯ 𝑏﷯ = 𝑖﷯ + 2 𝑗﷯ − 3 𝑘﷯ 𝑐﷯ = 3 𝑖﷯ − λ 𝑗﷯ + 5 𝑘﷯ 𝑎﷯ 𝑏﷯ 𝑐﷯﷯ = 2﷮−1﷮1﷮1﷮2﷮−3﷮3﷮−𝜆﷮5﷯﷯ 0 = 2 2×5﷯−(−λ×−3) ﷯ − (−1) 1×5﷯−(3×−3) ﷯ + 1 1×−λ﷯−(3×2) ﷯ 0 = 2 10−3λ﷯ + 5+9 ﷯ + −λ−6﷯ 0 = 20 – 6λ + 14 – λ +– 6 0 = –7λ + 28 7λ = 28 λ = 28﷮7﷯ 𝝀 = 4 Therefore, 𝑎﷯,𝑏, 𝑐﷯ are coplanar if λ = 4 Supplementary Exercise Q5 Find the value of 𝜆 if the following vectors are co-planar (ii) 𝑎﷯ = 2 𝑖﷯ + 𝑗﷯ + 𝑘﷯, 𝑏﷯ = 2 𝑖﷯ − 𝜆 𝑗﷯ + 𝑘﷯, and 𝑐﷯ = 5 𝑖﷯ − 𝑗﷯ − 3 𝑘﷯ Three vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ are coplanar if 𝒂﷯ 𝒃﷯ 𝒄﷯﷯ = 0 Given, 𝑎﷯ = 2 𝑖﷯ + 𝑗﷯ + 𝑘﷯ 𝑏﷯ = 2 𝑖﷯ − λ 𝑗﷯ + 𝑘﷯ 𝑐﷯ = 5 𝑖﷯ + 𝑗﷯ – 3 𝑘﷯ 𝑎﷯ 𝑏﷯ 𝑐﷯﷯ = 2﷮1﷮1﷮2﷮−𝜆﷮1﷮5﷮1﷮−3﷯﷯ 0 = 2 −λ×−3﷯−(1×1)﷯ − 1 2×−3﷯−(5×1)﷯ + 1 2×1﷯−(5×−λ) ﷯ 0 = 2 3λ−1﷯ − 1 −6−5﷯ + 1 2+5λ﷯ 0 = 6λ – 2 + 11 + 2 + 5λ 0 = 11λ + 11 –11λ = 11 λ = 11﷮−11﷯ 𝝀 = –1 Therefore, 𝑎﷯,𝑏, 𝑐﷯ are coplanar if λ = –1

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