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Supplementary Exercise Q10 - If vectors A, B, C are coplanar, prove - 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 Q10 If the vectors 𝐴﷯ = a 𝑖﷯ + 𝑗﷯ + 𝑘﷯, 𝐵﷯ = 𝑖﷯ + b 𝑗﷯ + 𝑘﷯ and 𝐶﷯ = 𝑖﷯ + 𝑗﷯ + c 𝑘﷯ are coplanar, then 1﷮1−𝑎﷯ + 1﷮1−𝑏﷯ + 1﷮1−𝑐﷯ = 1, where a, b, c ≠ 1 Since Vectors 𝐴﷯, 𝐵﷯, 𝐶﷯ are coplanar , ∴ 𝐴﷯ 𝐵﷯ 𝐶﷯﷯ = 0 𝐴﷯ 𝐵﷯ 𝐶﷯﷯ = 𝑎﷮1﷮1﷮1﷮𝑏﷮1﷮1﷮1﷮𝑐﷯﷯ = 0 C1 C1 C3 C2 C2 C3 𝑎−1﷮0﷮1﷮0﷮𝑏−1﷮1﷮1−𝑐﷮1−𝑐﷮𝑐﷯﷯ = 0 𝑎−1﷮0﷮1﷮0﷮𝑏−1﷮1﷮1−𝑐﷮1−𝑐﷮𝑐﷯﷯ = 0 Expanding Determinant along C3 1 0(1 − c) − (1 − c)(b − 1)﷯ −1 (a − 1) (1 − c) − 0﷯ + c (a − 1) (b − 1) − 0﷯ (1 − c) (1 − b) − (a − 1) (1 − c) + c (1− a) (1 − b) = 0 (1 − c) (1 − b) + (1 − c) (1 − a) + c (1 − a) (1− b) = 0 Dividing L.H.S and R.H.S by (1 − a) (1 − b) (1 − c) 1 − 𝑐﷯ 1 − 𝑏﷯ + 1 − 𝑐﷯ 1 − 𝑎﷯ + 𝑐 1 − 𝑎﷯ 1 − 𝑏﷯ ﷮ 1 − 𝑎﷯ 1 − 𝑏﷯ 1 −𝑐﷯﷯ =0 1﷮1 − 𝑎﷯+ 1﷮1 − 𝑏﷯+ 𝑐﷮1 − 𝑐﷯=0 1﷮1 − 𝑎﷯+ 1﷮1 − 𝑏﷯+ 𝑐﷮1 − 𝑐﷯=0 Adding 1 both sides 1﷮1 − 𝑎﷯+ 1﷮1 − 𝑏﷯+ 𝑐﷮1 − 𝑐﷯+1=1 1﷮1 − 𝑎﷯+ 1﷮1−𝑏﷯+ 𝑐 + 1 − 𝑐﷮1 − 𝑐﷯=1 1﷮1 − 𝑎﷯+ 1﷮1 − 𝑏﷯+ 1﷮1 − 𝑐﷯=1 Hence Proved

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