Examples 10 i.jpg

Examples 10 ii.jpg
Examples 10 iii.jpg

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

Transcript

Example 10 Find the angle between the pair of lines (๐‘ฅ + 3)/3 = (๐‘ฆ โˆ’ 1)/5 = (๐‘ง + 3)/4 and (๐‘ฅ + 1)/1 = (๐‘ฆ โˆ’ 4)/1 = (๐‘ง โˆ’ 5)/2 Angle between the pair of lines (๐‘ฅ โˆ’ ๐‘ฅ1)/๐‘Ž1 = (๐‘ฆ โˆ’ ๐‘ฆ1)/๐‘1 = (๐‘ง โˆ’ ๐‘ง1)/๐‘1 and (๐‘ฅ โˆ’ ๐‘ฅ2)/๐‘Ž2 = (๐‘ฆ โˆ’ ๐‘ฆ2)/๐‘2 = (๐‘ง โˆ’ ๐‘ง2)/๐‘2 is given by cos ฮธ = |(๐’‚_๐Ÿ ๐’‚_๐Ÿ + ๐’ƒ_๐Ÿ ๐’ƒ_๐Ÿ +ใ€– ๐’„ใ€—_๐Ÿ ๐’„_๐Ÿ)/(โˆš(ใ€–๐’‚_๐Ÿใ€—^๐Ÿ + ใ€–๐’ƒ_๐Ÿใ€—^๐Ÿ+ ใ€–๐’„_๐Ÿใ€—^๐Ÿ ) โˆš(ใ€–๐’‚_๐Ÿใ€—^๐Ÿ +ใ€–ใ€– ๐’ƒใ€—_๐Ÿใ€—^๐Ÿ+ ใ€–๐’„_๐Ÿใ€—^๐Ÿ ))| cos ฮธ = |(3.1 + 5.1 + 4.2)/(โˆš(3^2 + 5^2 + 4^2 ) ร— โˆš(1^2 + 1^2 + 2^2 ))| (๐’™ + ๐Ÿ‘)/๐Ÿ‘ = (๐’š โˆ’ ๐Ÿ)/๐Ÿ“ = (๐’› + ๐Ÿ‘)/๐Ÿ’ (๐‘ฅ โˆ’ (โˆ’3))/3 = (๐‘ฆ โˆ’ 1)/5 = (๐‘ง โˆ’ (โˆ’3))/4 Comparing with (๐‘ฅ โˆ’ ๐‘ฅ1)/๐‘Ž1 = (๐‘ฆ โˆ’ ๐‘ฆ1)/๐‘1 = (๐‘ง โˆ’ ๐‘ง1)/๐‘1 x1 = โˆ’3, y1 = 1, z1 = โ€“3 & ๐‘Ž1 = 3, b1 = 5, c1 = 4 (๐’™ + ๐Ÿ)/๐Ÿ = (๐’š โˆ’ ๐Ÿ’)/๐Ÿ = (๐’› โˆ’ ๐Ÿ“)/๐Ÿ (๐‘ฅ โˆ’ (โˆ’1))/1 = (๐‘ฆ โˆ’ 4)/1 = (๐‘ง โˆ’ 5)/2 Comparing with (๐‘ฅ โˆ’ ๐‘ฅ2)/๐‘Ž2 = (๐‘ฆ โˆ’ ๐‘ฆ2)/๐‘2 = (๐‘ง โˆ’ ๐‘ง2)/๐‘2 ๐‘ฅ2 = โˆ’1, y2 = 4, z2 = 5 & ๐‘Ž2 = 1, ๐‘2 = 1, ๐‘2 = 2 Now, cos ฮธ = |(๐‘Ž_1 ๐‘Ž_2 + ๐‘_1 ๐‘_2 +ใ€– ๐‘ใ€—_1 ๐‘_2)/(โˆš(ใ€–๐‘Ž_1ใ€—^2 + ใ€–๐‘_1ใ€—^2+ ใ€–๐‘_1ใ€—^2 ) โˆš(ใ€–๐‘Ž_2ใ€—^2 +ใ€–ใ€– ๐‘ใ€—_2ใ€—^2+ ใ€–๐‘_2ใ€—^2 ))| = |((3 ร— 1) + (5 ร— 1) + (4 ร— 2))/(โˆš(3^2 + 5^2 + 4^2 ) ร— โˆš(1^2 + 1^2 + 2^2 ))| = |(3 + 5 + 8)/(โˆš(9 + 25 + 16) โˆš(1 + 1 + 4))| = |16/(โˆš50 โˆš6)| = |16/(5โˆš2 ร— โˆš2 โˆš3)| = |16/(5 ร— 2 ร— โˆš3)| = 8/(5 โˆš3) = 8/(5 โˆš3) ร— โˆš3/โˆš3 = (8โˆš3)/(15 ) So, cos ฮธ = (8โˆš3)/(15 ) โˆด ฮธ = cos-1((๐Ÿ–โˆš๐Ÿ‘)/(๐Ÿ๐Ÿ“ )) Therefore, the angle between the given pair of line is cos-1 ((8โˆš3)/(15 ))

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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.