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Ex 10.5, 6 (Supplementary NCERT) - Find x such that A (3, 2, 1), B (4,

Ex 10.5, 6 (Supplementary NCERT) - Chapter 10 Class 12 Vector Algebra - Part 2

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Ex 10.5, 6 (Supplementary NCERT) Find x such that A (3, 2, 1), B = (4, x, 5), C(4, 2, –2) and D (6, 5, –1) are coplanar Four points A, B, C, D are coplanar if the three vectors (𝐴𝐡) βƒ— , (𝐴𝐢) βƒ— and (𝐴𝐷) βƒ— are coplanar. i.e. [(𝑨𝑩) βƒ—, (𝑨π‘ͺ) βƒ—, (𝑨𝑫) βƒ— ] = 0 A (3, 2, 1), B (4, x, 5) (𝑨𝑩) βƒ— = (4 – 3)𝑖 Μ‚ + (x βˆ’ 2)𝑗 Μ‚ + (5 – 1) π‘˜ Μ‚ = 1π’Š Μ‚ + (x βˆ’ 2)𝒋 Μ‚ + 4π’Œ Μ‚ A (3, 2, 1), C (4, 2, –2) (𝑨π‘ͺ) βƒ— = (4 – 3)𝑖 Μ‚ + (2 – 2)𝑗 Μ‚ + (–2 – 1) π‘˜ Μ‚ = 1π’Š Μ‚ + 0𝒋 Μ‚ βˆ’ 3π’Œ Μ‚ A (3, 2, 1), D (6, 5, –1) (𝑨𝑫) βƒ— = (6 – 3)𝑖 Μ‚ + (5 – 2)𝑗 Μ‚ + (–1 – 1) π‘˜ Μ‚ = 3π’Š Μ‚ + 3𝒋 Μ‚ – 2π’Œ Μ‚ Now, [(𝐴𝐡) βƒ—, (𝐴𝐢) βƒ—, (𝐴𝐷) βƒ— ] = |β– 8(1&(π‘₯βˆ’2)&4@1&0&βˆ’3@3&3&βˆ’2)| 0 = 1[(0Γ—βˆ’2)βˆ’(3Γ—βˆ’3) ] βˆ’ (x βˆ’ 2) [(1Γ—βˆ’2)βˆ’(3Γ—βˆ’3)] + 4[(1Γ—3)βˆ’(3Γ—0) ] 0 = 1[0βˆ’(βˆ’9)]βˆ’"(x βˆ’ 2)" [βˆ’2βˆ’(βˆ’9)]+4[3βˆ’0] 0 = 1(9) – "(x βˆ’ 2)"(–2 + 9) + 4(3) 0 = 9 – (x – 2) (7) + 12 0 = 9 – 7x + 14 + 12 0 = 35 – 7x 7x = 35 x = 35/7 x = 5

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