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Last updated at Feb. 1, 2020 by Teachoo

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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 ๐1, ๐1, c1 and ๐_2 , ๐_2, ๐_2 is given by cos ฮธ = |(๐_๐ ๐_๐ + ๐_๐ ๐_๐ + ๐๐๐๐)/(โ(ใ๐_๐ใ^๐ + ใ๐_๐ใ^๐+ใ ๐_๐ใ^๐ ) โ(ใ๐_๐ใ^๐ + ใ๐_๐ใ^๐+ใ ๐_๐ใ^๐ ))| A line passing through A (๐ฅ_1, ๐ฆ_1, ๐ง_1) and B (๐ฅ_2, ๐ฆ_2, ๐ง_2) has direction ratios (๐ฅ_2 โ ๐ฅ_1), (๐ฆ_2 โ ๐ฆ_1), (๐ง_2 โ ๐ง_1) AB A (1, 2, 3) , B (4, 5, 7) Direction ratios of AB (4 โ 1), (5 โ 2),(7 โ 3) = 3, 3, 4 โด ๐1 = 3, ๐1 = 3, ๐1 = 4 CD C (โ4, 3, โ6) ,D (2, 9, 2) Direction ratios of CD (2 โ (โ4)), (9 โ 3),(2 โ (โ6)) = 6, 6, 8 โด ๐2 = 6, ๐2 = 6, ๐2 = 8 Now, cos ฮธ = |(๐_1 ๐_2 + ๐_1 ๐_2 + ๐1๐2)/(โ(ใ๐_1ใ^2 + ใ๐_1ใ^2+ใ ๐_1ใ^2 ) โ(ใ๐_2ใ^2 + ใ๐_2ใ^2+ใ ๐_2ใ^2 ))| cos ฮธ = |(3 ร 6 + 3 ร 6 + 4 ร 8 )/(โ(32 + 32 + 42) โ(62 + 62 + 82))| = |(18 + 18 + 32 )/(โ(9 + 9 + 16) โ(36 + 36 + 64))| = |68/(โ34 โ136)| = |68/(โ34 โ(4 ร 34))| = |68/(โ34 ร โ4 ร โ34)| = |68/(โ34 ร โ34ร โ4)| = |68/(34 ร 2 )| = |68/68| = 1 โด cos ฮธ = 1 So, ฮธ = 0ยฐ Therefore, angle between AB and CD is 0ยฐ .

Miscellaneous

Misc 1
Important

Misc 2

Misc 3

Misc 4 Important

Misc 5 Important You are here

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc 19 Important

Misc 20 Important

Misc 21 Important

Misc 22 Important

Misc 23 Important

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

About the Author

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.