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

Transcript

Misc 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (โ€“ 4, 3, โ€“ 6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.Angle between a pair of lines having direction ratios ๐‘Ž1, ๐‘1, c1 and ๐‘Ž_2 , ๐‘_2, ๐‘_2 is given by cos ฮธ = |(๐’‚_๐Ÿ ๐’‚_๐Ÿ + ๐’ƒ_๐Ÿ ๐’ƒ_๐Ÿ + ๐’„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€–๐’ƒ_๐Ÿใ€—^๐Ÿ+ใ€– ๐’„_๐Ÿใ€—^๐Ÿ ))| A line passing through A (๐‘ฅ_1, ๐‘ฆ_1, ๐‘ง_1) and B (๐‘ฅ_2, ๐‘ฆ_2, ๐‘ง_2) has direction ratios (๐‘ฅ_2 โˆ’ ๐‘ฅ_1), (๐‘ฆ_2 โˆ’ ๐‘ฆ_1), (๐‘ง_2 โˆ’ ๐‘ง_1) AB A (1, 2, 3) , B (4, 5, 7) Direction ratios of AB (4 โˆ’ 1), (5 โˆ’ 2),(7 โˆ’ 3) = 3, 3, 4 โˆด ๐’‚1 = 3, ๐’ƒ1 = 3, ๐’„1 = 4 CD C (โˆ’4, 3, โˆ’6) ,D (2, 9, 2) Direction ratios of CD (2 โˆ’ (โ€“4)), (9 โˆ’ 3),(2 โ€“ (โ€“6)) = 6, 6, 8 โˆด ๐’‚2 = 6, ๐’ƒ2 = 6, ๐’„2 = 8 Now, cos ฮธ = |(๐‘Ž_1 ๐‘Ž_2 + ๐‘_1 ๐‘_2 + ๐‘1๐‘2)/(โˆš(ใ€–๐‘Ž_1ใ€—^2 + ใ€–๐‘_1ใ€—^2+ใ€– ๐‘_1ใ€—^2 ) โˆš(ใ€–๐‘Ž_2ใ€—^2 + ใ€–๐‘_2ใ€—^2+ใ€– ๐‘_2ใ€—^2 ))| cos ฮธ = |(3 ร— 6 + 3 ร— 6 + 4 ร— 8 )/(โˆš(32 + 32 + 42) โˆš(62 + 62 + 82))| = |(18 + 18 + 32 )/(โˆš(9 + 9 + 16) โˆš(36 + 36 + 64))| = |68/(โˆš34 โˆš136)| = |68/(โˆš34 โˆš(4 ร— 34))| = |68/(โˆš34 ร— โˆš4 ร— โˆš34)| = |68/(โˆš34 ร— โˆš34ร— โˆš4)| = |68/(34 ร— 2 )| = |68/68| = 1 โˆด cos ฮธ = 1 So, ฮธ = 0ยฐ Therefore, angle between AB and CD is 0ยฐ .

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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.