Ex 10.4, 7 - Show that a x (b + c) =  a x b + a x c - Ex 10.4

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Ex 10.4, 7 - Chapter 10 Class 12 Vector Algebra - Part 2

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Ex 10.4, 7 - Chapter 10 Class 12 Vector Algebra - Part 3

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

Transcript

Ex 10.4, 7 Let the vectors ๐‘Ž โƒ— ๐‘ โƒ—, ๐‘ โƒ— be given as ๐‘Ž_1 ๐‘– ฬ‚ + ๐‘Ž_2 ๐‘— ฬ‚ +๐‘Ž_3 ๐‘˜ ฬ‚, ๐‘_1 ๐‘– ฬ‚ + ๐‘_2 ๐‘— ฬ‚ +๐‘_3 ๐‘˜ ฬ‚, ๐‘_1 ๐‘– ฬ‚ + ๐‘_2 ๐‘— ฬ‚ +๐‘_3 ๐‘˜ ฬ‚ Then show that ๐‘Ž โƒ— ร— (๐‘ โƒ— + ๐‘ โƒ—) =๐‘Ž โƒ— ร—๐‘ โƒ— + ๐‘Ž โƒ— ร— ๐‘ โƒ—. Let ๐‘Ž โƒ— = ๐‘Ž_1 ๐‘– ฬ‚ + ๐‘Ž_2 ๐‘— ฬ‚ +๐‘Ž_3 ๐‘˜ ฬ‚ ๐‘ โƒ— = ๐‘_1 ๐‘– ฬ‚ + ๐‘_2 ๐‘— ฬ‚ +๐‘_3 ๐‘˜ ฬ‚, ๐‘ โƒ— = ๐‘_1 ๐‘– ฬ‚ + ๐‘_2 ๐‘— ฬ‚ + ๐‘_3 ๐‘˜ ฬ‚ We need to show : ๐‘Ž โƒ— ร— (๐‘ โƒ— + ๐‘ โƒ—) = ๐‘Ž โƒ— ร— ๐‘ โƒ— + ๐‘Ž โƒ— ร— ๐‘ โƒ— RHS (๐‘Ž โƒ— ร— ๐‘ โƒ—) = |โ– 8(๐‘– ฬ‚&๐‘— ฬ‚&๐‘˜ ฬ‚@๐‘Ž1&๐‘Ž2&๐‘Ž3@๐‘1&๐‘2&๐‘3)| = (๐‘Ž2 ๐‘3 โˆ’ ๐‘2 ๐‘Ž3) ๐‘– ฬ‚ โˆ’ (๐‘Ž1 ๐‘3 โˆ’ ๐‘1 ๐‘Ž3) ๐‘— ฬ‚ + (๐‘Ž1 ๐‘2 โˆ’ ๐‘1 ๐‘Ž2) ๐‘˜ ฬ‚ (๐‘Ž โƒ— ร— ๐‘ โƒ—) = |โ– 8(๐‘– ฬ‚&๐‘— ฬ‚&๐‘˜ ฬ‚@๐‘Ž1&๐‘Ž2&๐‘Ž3@๐‘1&๐‘2&๐‘3)| = (๐‘Ž2" " ๐‘3 โˆ’ ๐‘2" " ๐‘Ž3) ๐‘– ฬ‚ โˆ’ (๐‘Ž1" " ๐‘3 โˆ’ ๐‘1" " ๐‘Ž3) ๐‘— ฬ‚ + (๐‘Ž1" " ๐‘2 โˆ’ ๐‘1" " ๐‘Ž2) ๐‘˜ ฬ‚ (๐’‚ โƒ— ร— ๐’ƒ โƒ—) + (๐’‚ โƒ— ร— ๐’„ โƒ—) = [๐‘Ž2" " ๐‘3โˆ’๐‘2 ๐‘Ž3+๐‘Ž2 ๐‘3โˆ’๐‘2๐‘Ž3] ๐‘– ฬ‚ โˆ’ [๐‘Ž1" " ๐‘3โˆ’๐‘1 ๐‘Ž3+๐‘Ž1 ๐‘3โˆ’๐‘1๐‘Ž3] ๐‘— ฬ‚ + [๐‘Ž1" " ๐‘2โˆ’๐‘1 ๐‘Ž2+๐‘Ž1 ๐‘2โˆ’๐‘1๐‘Ž2] ๐‘˜ ฬ‚ Since the corresponding components are equal, So, ๐‘Ž โƒ— ร— (๐‘ โƒ— + ๐‘ โƒ—) = ๐‘Ž โƒ— ร— ๐‘ โƒ— + ๐‘Ž โƒ— ร— ๐‘ โƒ— Hence proved. Ex 10.4, 7 Let the vectors , be given as 1 + 2 + 3 , 1 + 2 + 3 , 1 + 2 + 3 Then show that ( + ) = + . Let = 1 + 2 + 3 = 1 + 2 + 3 , = 1 + 2 + 3 We need to show : ( + ) = + LHS ( + ) = ( 1 + 1) + ( 2 + 2) + ( 3 + 3) ( + ) = 1 ( 1+ 1) 2 ( 2+ 2) 3 ( 3+ 3) = 2 ( 3 + 3) ( 2 + 2) 3 1 ( 3 + 3) ( 1 + 1) 3 + 1 ( 2 + 2) ( 1 + 1) 2 = 2 3+ 2 3 2 3 2 3 1 3 1 3+ 1 3 1 3 + 1 2+ 1 2 1 2 1 2

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.