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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Ex 10.5, 3 (Supplementary NCERT) Find ๐œ† if the vectors ๐‘–ย ฬ‚ โ€“ ๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚, 3๐‘–ย ฬ‚ + ๐‘—ย ฬ‚ + 2๐‘˜ย ฬ‚ and ๐‘–ย ฬ‚ + ๐œ†๐‘—ย ฬ‚ โ€“ 3๐‘˜ย ฬ‚ are coplanar.Three vectors ๐‘Žย โƒ—, ๐‘ย โƒ—, ๐‘ย โƒ— are coplanar if [๐’‚ย โƒ—" " ๐’ƒย โƒ—" " ๐’„ย โƒ— ] = 0 Let ๐‘Žย โƒ— = ๐‘–ย ฬ‚ โ€“ ๐‘—ย ฬ‚ + ๐‘˜ย ฬ‚ ๐‘ย โƒ— = 3๐‘–ย ฬ‚ + ๐‘—ย ฬ‚ + 2๐‘˜ย ฬ‚ ๐‘ย โƒ— = ๐‘–ย ฬ‚ + ๐œ†๐‘—ย ฬ‚ โ€“ 3๐‘˜ย ฬ‚ [๐‘Žย โƒ—" " ๐‘ย โƒ—" " ๐‘ย โƒ— ] = |โ– 8(1&โˆ’1&1@3&1&2@1&"๐œ†" &โˆ’3)| 0 = 1[(1ร—โˆ’3)โˆ’("๐œ†" ร—2) ] โˆ’ (โ€“1)[(3ร—โˆ’3)โˆ’(1ร—2) ] + 1[(3ร—"๐œ†" )โˆ’(1ร—1) ] 0 = 1 [โˆ’3โˆ’2"๐œ†" ]+1[โˆ’9โˆ’2]+1[3"๐œ†" โˆ’1] 0 = โ€“3 โ€“ 2๐œ† โ€“ 11 + 3๐œ† โ€“ 1 0 = โ€“3 โ€“ 11 โ€“ 1 + 3๐œ† โ€“ 2ฮป 0 = โ€“15 + ๐œ† 15 = ฮป ๐€ = 15 Therefore, ๐‘Žย โƒ—,๐‘,๐‘ย โƒ— are coplanar if ฮป = 15

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