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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Example 29 (Supplementary NCERT) Show that the four points A, B, C and D with position vectors 4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚, โˆ’(๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚), 3๐‘–ย ฬ‚ + 9๐‘—ย ฬ‚ + 4๐‘˜ย ฬ‚ and โˆ’4๐‘–ย ฬ‚ + 4๐‘—ย ฬ‚ + 4๐‘˜ย ฬ‚, respectively are coplanar Four points A, B, C, D are coplanar if the three vectors (๐ด๐ต)ย โƒ— , (๐ด๐ถ)ย โƒ— and (๐ด๐ท)ย โƒ— are coplanar. i.e. [(๐‘จ๐‘ฉ)ย โƒ—, (๐‘จ๐‘ช)ย โƒ—, (๐‘จ๐‘ซ)ย โƒ— ] = 0 A (4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚) B (โˆ’๐‘—ย ฬ‚ โˆ’ ๐‘˜ย ฬ‚) (๐‘จ๐‘ฉ)ย โƒ— = (0๐‘–ย ฬ‚ โˆ’ ๐‘—ย ฬ‚ โˆ’ ๐‘˜ย ฬ‚) โˆ’ (4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚) = โˆ’4๐’Šย ฬ‚ โˆ’ 6๐’‹ย ฬ‚ โˆ’ 2๐’Œย ฬ‚ A (4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚) C (3๐‘–ย ฬ‚ + 9๐‘—ย ฬ‚ + 4๐‘˜ย ฬ‚) (๐‘จ๐‘ช)ย โƒ— = (3๐‘–ย ฬ‚ + 9๐‘—ย ฬ‚ + 4๐‘˜ย ฬ‚) โˆ’ (4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚) = โ€“๐’Šย ฬ‚ + 4๐’‹ย ฬ‚ + 3๐’Œย ฬ‚ A (4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚) D (โˆ’4๐‘–ย ฬ‚ + 4๐‘—ย ฬ‚ + 4๐‘˜ย ฬ‚) (๐‘จ๐‘ซ)ย โƒ— = (โˆ’4๐‘–ย ฬ‚ + 4๐‘—ย ฬ‚ + 4๐‘˜ย ฬ‚) โˆ’ (4๐‘–ย ฬ‚ + 5๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚) = โ€“8๐’Šย ฬ‚ โˆ’ ๐’‹ย ฬ‚ + 3๐’Œย ฬ‚ [(๐ด๐ต)ย โƒ—, (๐ด๐ถ)ย โƒ—, (๐ด๐ท)ย โƒ— ] = |โ– 8(โˆ’4&โˆ’6&โˆ’2@โˆ’1&4&3@โˆ’8&โˆ’1&3)| = โˆ’4[(4ร—3)โˆ’(โˆ’1ร—3) ] โˆ’ (โˆ’6) [(โˆ’1ร—3)โˆ’(โˆ’8ร—3)] + (โˆ’2)[(โˆ’1ร—โˆ’1)โˆ’(โˆ’8ร—4) ] = โ€“4 [12+3]+6[โˆ’3+24]โˆ’2[1+32] = โˆ’4 (15) + 6 (21) โˆ’ 2 (33) = โˆ’60 + 126 โˆ’ 66 = โˆ’126+ 126 = 0 โˆด[(๐ด๐ต)ย โƒ—, (๐ด๐ถ)ย โƒ—, (๐ด๐ท)ย โƒ— ] = 0 Therefore, points A, B, C and D are coplanar.

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