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

Transcript

Ex 10.3, 5 Show that each of the given three vectors is a unit vector: 1/7 (2๐‘– ฬ‚ + 3๐‘— ฬ‚ + 6๐‘˜ ฬ‚), 1/7 (3๐‘– ฬ‚ โ€“ 6๐‘— ฬ‚ + 2๐‘˜ ฬ‚), 1/7 (6๐‘– ฬ‚ + 2๐‘— ฬ‚ โ€“ 3๐‘˜ ฬ‚), Also, show that they are mutually perpendicular to each other. ๐‘Ž โƒ— = 1/7 (2๐‘– ฬ‚ + 3๐‘— ฬ‚ + 6๐‘˜ ฬ‚) = 2/7 ๐‘– ฬ‚ + 3/7 ๐‘— ฬ‚ + 6/7 ๐‘˜ ฬ‚ ๐‘ โƒ— = 1/7 (3๐‘– ฬ‚ โˆ’ 6๐‘— ฬ‚ + 2๐‘˜ ฬ‚) = 3/7 ๐‘– ฬ‚ โ€“ 6/7 ๐‘— ฬ‚ + 2๐‘˜/7 ๐‘˜ ฬ‚ ๐‘ โƒ— = 1/7 (6๐‘– ฬ‚ + 2๐‘— ฬ‚ - 3๐‘˜ ฬ‚) = 6/7 ๐‘– ฬ‚ + 2/7 ๐‘— ฬ‚ โ€“ 3/7 ๐‘˜ ฬ‚ Magnitude of ๐‘Ž โƒ— = โˆš((2/7)^2+(3/7)^2+(6/7)^2 ) |๐‘Ž โƒ— | = โˆš(4/49+9/49+36/49) = โˆš(49/49) = 1 Since |๐‘Ž โƒ— | = 1 So, ๐‘Ž โƒ— is a unit vector. Magnitude of ๐‘ โƒ— = โˆš((3/7)^2+((โˆ’6)/7)^2+(2/7)^2 ) |๐‘ โƒ— | = โˆš(9/49+36/49+4/49)= โˆš(49/49) = 1 Since |๐‘ โƒ— | = 1 So, ๐‘ โƒ— is a unit vector. Magnitude of ๐‘ โƒ— = โˆš((6/7)^2+(2/7)^2+((โˆ’3)/7)^2 ) |๐‘ โƒ— | = โˆš(36/49+4/49+9/49) = โˆš(49/49) = 1 Since |๐‘ โƒ— | = 1, So, ๐‘ โƒ— is a unit vector Now, we need to show that they are mutually perpendicular to each other. So, ๐’‚ โƒ—. ๐’ƒ โƒ— = ๐’ƒ โƒ—. ๐’„ โƒ— = ๐’„ โƒ— . ๐’‚ โƒ— = 0 Thus, they are mutually perpendiculars to each other. ๐‘Ž โƒ— = 2/7 ๐‘– ฬ‚ + 3/7 ๐‘— ฬ‚ + 6/7 ๐‘˜ ฬ‚ ๐‘ โƒ— = 3/7 ๐‘– ฬ‚ โ€“ 6/7 ๐‘— ฬ‚ + 2/7 ๐‘˜ ฬ‚ ๐’‚ โƒ—. ๐’ƒ โƒ— = 2/7 . 3/7 + 3/7 (โˆ’6/7) + 6/7 . 2/7 = 6/49 โˆ’ 18/49 + 12/49 = 0 ๐‘ โƒ— = 3/7 ๐‘– ฬ‚ โˆ’ 6/7 ๐‘— ฬ‚ + 2/7 ๐‘˜ ฬ‚ ๐‘ โƒ— = 6/7 ๐‘– ฬ‚ + 2/7 ๐‘— ฬ‚ โ€“ 3/7 ๐‘˜ ฬ‚ ๐’ƒ โƒ—. ๐’„ โƒ— = 3/7 . 6/7 + (โˆ’6/7) 2/7 + 2/7 . ((โˆ’3)/7) = 18/49 โˆ’ 12/49 โˆ’ 6/49 = 0 ๐‘ โƒ— = 6/7 ๐‘– ฬ‚ + 2/7 ๐‘— ฬ‚ โˆ’ 3/7 ๐‘˜ ฬ‚ ๐‘Ž โƒ— = 2/7 ๐‘– ฬ‚ + 3/7 ๐‘— ฬ‚ + 6/7 ๐‘˜ ฬ‚ ๐’„ โƒ—. ๐’‚ โƒ— = 6/7 . 2/7 + 2/7. 3/7 + ((โˆ’3)/7) 6/7 = 12/49 + 6/49 โˆ’ 18/49 = 0

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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.