Question 10 (OR 2 nd question)

Find the angle between the vectors:

  a = i + j – k and b = i – j + k

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  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Transcript

Question 10 (OR 2nd question) Find the angle between the vectors: ๐‘Ž โƒ— = ๐‘– ฬ‚ + ๐‘— ฬ‚ โ€“ ๐‘˜ ฬ‚ and ๐‘ โƒ— = ๐‘– ฬ‚ โ€“ ๐‘— ฬ‚ + ๐‘˜ ฬ‚ Given ๐‘Ž โƒ— = ๐‘– ฬ‚ + ๐‘— ฬ‚ โ€“ ๐‘˜ ฬ‚ and ๐‘ โƒ— = ๐‘– ฬ‚ โ€“ ๐‘— ฬ‚ + ๐‘˜ ฬ‚ We know that ๐‘Ž โƒ— . ๐‘ โƒ— = |๐‘Ž โƒ— ||๐‘ โƒ— | cos ฮธ where ฮธ is the angle between ๐‘Ž โƒ— & ๐‘ โƒ— Now, ๐’‚ โƒ—. ๐’ƒ โƒ— = (๐‘– ฬ‚ + ๐‘— ฬ‚ โ€“ ๐‘˜ ฬ‚). (๐‘– ฬ‚ โ€“ ๐‘— ฬ‚ + ๐‘˜ ฬ‚) = (1๐‘– ฬ‚ + 1๐‘— ฬ‚ โ€“ ๐‘˜ ฬ‚). (1๐‘– ฬ‚ โˆ’ 1๐‘— ฬ‚ + 1๐‘˜ ฬ‚) = 1 ร— 1 + 1 ร— (โˆ’1) + (-1) ร— 1 = 1 โ€“ 1 โ€“ 1 = โ€“1 Magnitude of ๐‘Ž โƒ— = โˆš(12+12+(โˆ’1)2) |๐‘Ž โƒ— |= โˆš(1+1+1) = โˆš3 Magnitude of ๐‘ โƒ— = โˆš(12+(โˆ’1)2+12) |๐‘ โƒ— |= โˆš(1+1+1) = โˆš3 Now, ๐‘Ž โƒ— . ๐‘ โƒ— = |๐‘Ž โƒ— ||๐‘ โƒ— | cos ฮธ โ€“1 = โˆš3 ร— โˆš3 ร— cos ฮธ โ€“1 = 3 cos ฮธ cos ฮธ = (โˆ’1)/3 ฮธ = ๐’„๐’๐’”^(โˆ’๐Ÿ) ((โˆ’๐Ÿ)/๐Ÿ‘) Thus, the angle between ๐‘Ž โƒ— and ๐‘ โƒ— is cos-1((โˆ’๐Ÿ)/๐Ÿ‘)

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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.