Ex 11.2, 10 - Class 12th - 3D Geometry - Find angle between

Ex 11.2, 10 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Ex 11.2, 10 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

 

 

 

 

  1. Chapter 11 Class 12 Three Dimensional Geometry (Term 2)
  2. Serial order wise

Transcript

Ex 11.2, 10 Find the angle between the following pairs of lines: (i) ๐‘Ÿ โƒ— = 2๐‘– ฬ‚โˆ’ 5๐‘— ฬ‚ + ๐‘˜ ฬ‚ + ๐œ† (3๐‘– ฬ‚ + 2๐‘— ฬ‚ + 6๐‘˜ ฬ‚) and ๐‘Ÿ โƒ— = 7๐‘– ฬ‚ โ€“ 6๐‘˜ ฬ‚ + ๐œ‡(๐‘– ฬ‚ + 2๐‘— ฬ‚ + 2๐‘˜ ฬ‚) Ex 11.2, 10 Find the angle between the following pairs of lines: (i) ๐‘Ÿ โƒ— = 2๐‘– ฬ‚โˆ’ 5๐‘— ฬ‚ + ๐‘˜ ฬ‚ + ๐œ† (3๐‘– ฬ‚ + 2๐‘— ฬ‚ + 6๐‘˜ ฬ‚) and ๐‘Ÿ โƒ— = 7๐‘– ฬ‚ โ€“ 6๐‘˜ ฬ‚ + ๐œ‡(๐‘– ฬ‚ + 2๐‘— ฬ‚ + 2๐‘˜ ฬ‚) Angle between two vectors ๐‘Ÿ โƒ— = (๐‘Ž1) โƒ— + ๐œ† (๐‘1) โƒ— & ๐‘Ÿ โƒ— = (๐‘Ž2) โƒ— + ๐œ‡ (๐‘2) โƒ— is given by cos ฮธ = |((๐’ƒ๐Ÿ) โƒ— . (๐’ƒ๐Ÿ) โƒ—)/|(๐’ƒ๐Ÿ) โƒ— ||(๐’ƒ๐Ÿ) โƒ— | | Given, the pair of lines is ๐’“ โƒ— = (2๐’Š ฬ‚ โˆ’ 5๐’‹ ฬ‚ + ๐’Œ ฬ‚) + ๐œ† (3๐’Š ฬ‚ + 2๐’‹ ฬ‚ + 6๐’Œ ฬ‚) So, (๐‘Ž1) โƒ— = 2๐‘– ฬ‚ โˆ’ 5๐‘— ฬ‚ + 1๐‘˜ ฬ‚ (๐‘1) โƒ— = 3๐‘– ฬ‚ + 2๐‘— ฬ‚ + 6๐‘˜ ฬ‚ ๐’“ โƒ— = (7๐’Š ฬ‚ โˆ’ 6๐’Œ ฬ‚) + ๐ (๐’Š ฬ‚ + 2๐’‹ ฬ‚ + 2๐’Œ ฬ‚) So, (๐‘Ž2) โƒ— = 7๐‘– ฬ‚ + 0๐‘— ฬ‚ โˆ’ 6๐‘˜ ฬ‚ (๐‘2) โƒ— = 1๐‘– ฬ‚ + 2๐‘— ฬ‚ + 2๐‘˜ ฬ‚ Now, (๐‘1) โƒ—.(๐‘2) โƒ— = (3๐‘– ฬ‚ + 2๐‘— ฬ‚ + 6๐‘˜ ฬ‚) . (1๐‘– ฬ‚ + 2๐‘— ฬ‚ + 2๐‘˜ ฬ‚) = (3 ร— 1) + (2 ร— 2) + (6 ร— 2) = 3 + 4 + 12 = 19 Magnitude of (๐‘1) โƒ— = โˆš(32 + 22 + 62) |(๐‘1) โƒ— | = โˆš(9 + 4 + 36) = โˆš49 = 7 Magnitude of (๐‘2) โƒ— = โˆš(12+22+22) |(๐‘2) โƒ— | = โˆš(1+4+4) = โˆš9 = 3 Now, cos ฮธ = |((๐‘1) โƒ—.(๐‘2) โƒ—)/|(๐‘1) โƒ— ||(๐‘2) โƒ— | | cos ฮธ = |19/(7 ร— 3 )| cos ฮธ = 19/(21 ) โˆด ฮธ = cosโˆ’1 (๐Ÿ๐Ÿ—/(๐Ÿ๐Ÿ )) Therefore, the angle between the given vectors is cos โˆ’1(19/(21 ))

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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.