Ex 10.4, 8 - If either a = 0 or b = 0, then a x b = 0. Converse

Ex 10.4, 8 - Chapter 10 Class 12 Vector Algebra - Part 2

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

Transcript

Ex 10.4, 8 If either ๐‘Ž โƒ— = 0 โƒ— or ๐‘ โƒ— = 0 โƒ—, then ๐‘Ž โƒ— ร— ๐‘ โƒ— = 0 โƒ— . Is the converse true? Justify your answer with an example. Converse : If ๐‘Ž โƒ— ร— ๐‘ โƒ— = 0 โƒ—, then either ๐‘Ž โƒ— = 0 โƒ— or ๐‘ โƒ— = 0 โƒ— ๐‘Ž โƒ— ร— ๐‘ โƒ— = |๐‘Ž โƒ— ||๐‘ โƒ— | sin ฮธ ๐‘› ฬ‚ where, ฮธ = angle between ๐‘Ž โƒ— and ๐‘ โƒ— ๐‘› ฬ‚ = unit vector perpendicular to ๐‘Ž โƒ— ๐œ€ ๐‘ โƒ— Let ๐‘Ž โƒ— = 1๐‘– ฬ‚ + 1๐‘— ฬ‚ + 1๐‘˜ ฬ‚ & ๐‘ โƒ— = 2๐‘– ฬ‚ + 2๐‘— ฬ‚ + 2๐‘˜ ฬ‚ ๐‘Ž โƒ— ร— ๐‘ โƒ— = |โ– 8(๐‘– ฬ‚&๐‘— ฬ‚&๐‘˜ ฬ‚@1&1&1@2&2&2)| = ๐‘– ฬ‚ (1 ร— 2 โˆ’ 2 ร— 1) โˆ’ ๐‘— ฬ‚ (1 ร— 2 โˆ’2 ร— 1) + ๐‘˜ ฬ‚ (1 ร— 2 โˆ’ 2 ร— 1) + ๐‘˜ ฬ‚(1 ร— 2 โˆ’ 2 ร— 1) = ๐‘– ฬ‚ (2 โˆ’ 2) โˆ’ ๐‘— ฬ‚ (2 โˆ’2) + ๐‘˜ ฬ‚ (2 โˆ’ 2) = 0๐‘– ฬ‚ โˆ’ 0๐‘— ฬ‚ + 0๐‘˜ ฬ‚ = 0 โƒ— Here, ๐‘Ž โƒ— โ‰  0 โƒ— & ๐‘ โƒ—โ‰  0 โƒ— But ๐‘Ž โƒ— ร— ๐‘ โƒ— = 0 โƒ— Therefore, converse is not true.

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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.