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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise

Transcript

Example 29 (Supplementary NCERT) Show that the four points A, B, C and D with position vectors 4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚, โˆ’(๐‘— ฬ‚ + ๐‘˜ ฬ‚), 3๐‘– ฬ‚ + 9๐‘— ฬ‚ + 4๐‘˜ ฬ‚ & โˆ’4๐‘– ฬ‚ + 4๐‘— ฬ‚ + 4๐‘˜ ฬ‚, respectively coplanar Four points A, B, C, D are coplanar if the three vectors (๐ด๐ต) โƒ— , (๐ด๐ถ) โƒ— and (๐ด๐ท) โƒ— are coplanar. i.e. [(๐‘จ๐‘ฉ) โƒ—, (๐‘จ๐‘ช) โƒ—, (๐‘จ๐‘ซ) โƒ— ] = 0 A (4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚) B (โˆ’๐‘— ฬ‚ โˆ’ ๐‘˜ ฬ‚) (๐‘จ๐‘ฉ) โƒ— = (0๐‘– ฬ‚ โˆ’ ๐‘— ฬ‚ โˆ’ ๐‘˜ ฬ‚) โˆ’ (4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚) = โˆ’4๐’Š ฬ‚ โˆ’ 6๐’‹ ฬ‚ โˆ’ 2๐’Œ ฬ‚ A (4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚) C (3๐‘– ฬ‚ + 9๐‘— ฬ‚ + 4๐‘˜ ฬ‚) (๐‘จ๐‘ช) โƒ— = (3๐‘– ฬ‚ + 9๐‘— ฬ‚ + 4๐‘˜ ฬ‚) โˆ’ (4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚) = โ€“๐’Š ฬ‚ + 4๐’‹ ฬ‚ + 3๐’Œ ฬ‚ A (4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚) D (โˆ’4๐‘– ฬ‚ + 4๐‘— ฬ‚ + 4๐‘˜ ฬ‚) (๐‘จ๐‘ซ) โƒ— = (โˆ’4๐‘– ฬ‚ + 4๐‘— ฬ‚ + 4๐‘˜ ฬ‚) โˆ’ (4๐‘– ฬ‚ + 5๐‘— ฬ‚ + ๐‘˜ ฬ‚) = โ€“8๐’Š ฬ‚ โˆ’ ๐’‹ ฬ‚ + 3๐’Œ ฬ‚ [(๐ด๐ต) โƒ—, (๐ด๐ถ) โƒ—, (๐ด๐ท) 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) โƒ—, (๐ด๐ถ) โƒ—, (๐ด๐ท) โƒ— ] = 0 Therefore, points A, B, C and D are coplanar.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.