Show that the vectors 2i - 3j + 4k and -4i + 6j - 8k are collinear

Ex 10.2, 11 - Chapter 10 Class 12 Vector Algebra - Part 2
Ex 10.2, 11 - Chapter 10 Class 12 Vector Algebra - Part 3

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

Transcript

Ex 10.2, 11 (Method 1) Show that the vectors 2๐‘– ฬ‚ โˆ’ 3๐‘— ฬ‚ + 4๐‘˜ ฬ‚ and โˆ’ 4๐‘– ฬ‚ + 6 ๐‘— ฬ‚ โˆ’ 8๐‘˜ ฬ‚ are collinear.Two vectors are collinear if they are parallel to the same line. Let ๐‘Ž โƒ— = 2๐‘– ฬ‚ โˆ’ 3๐‘— ฬ‚ + 4๐‘˜ ฬ‚ and ๐‘ โƒ— = โ€“4๐‘– ฬ‚ + 6๐‘— ฬ‚ โ€“ 8๐‘˜ ฬ‚ Magnitude of ๐‘Ž โƒ— = โˆš(22+(โˆ’3)2+42) |๐‘Ž โƒ— | = โˆš(4+9+16) = โˆš29 Directions cosines of ๐‘Ž โƒ— = (2/โˆš29,(โˆ’3)/โˆš29,4/โˆš29) Magnitude of ๐‘ โƒ— =โˆš((โˆ’4)2+62+(โˆ’8)2) |๐‘ โƒ— | = โˆš(16+36+64) = โˆš116 = 2โˆš29 Directions cosines of ๐‘ โƒ— = ((โˆ’4)/(2โˆš29),6/(2โˆš29),(โˆ’8)/(2โˆš29)) = ((โˆ’2)/โˆš29,3/โˆš29,(โˆ’4)/โˆš29) = โˆ’1(2/โˆš29,(โˆ’3)/โˆš29,4/โˆš29) Hence, Direction cosines of ๐’‚ โƒ— = (โˆ’1) ร— Direction cosines of ๐’ƒ โƒ— โˆด They have opposite directions Since ๐‘Ž โƒ— and ๐‘ โƒ— are parallel to the same line ๐‘š โƒ—, they are collinear. Hence proved Ex 10.2, 11 (Method 2) Show that the vectors 2๐‘– ฬ‚ โˆ’ 3๐‘— ฬ‚ + 4๐‘˜ ฬ‚ and โˆ’ 4๐‘– ฬ‚ + 6 ๐‘— ฬ‚ โˆ’ 8๐‘˜ ฬ‚ are collinear.๐‘Ž โƒ— = 2๐‘– ฬ‚ โˆ’ 3๐‘— ฬ‚ + 4๐‘˜ ฬ‚ ๐‘ โƒ— = โ€“4๐‘– ฬ‚ + 6๐‘— ฬ‚ โ€“ 8๐‘˜ ฬ‚ Two vectors are collinear if their directions ratios are proportional ๐‘Ž_1/๐‘_1 = ๐‘Ž_2/๐‘_2 = ๐‘_3/๐‘_3 = ๐œ† 2/(โˆ’4) = (โˆ’3)/6 = 4/(โˆ’8) = (โˆ’1)/2 Since, directions ratios are proportional Hence, ๐‘Ž โƒ— & ๐‘ โƒ— are collinear

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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.