Last updated at Feb. 1, 2020 by Teachoo
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 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 .
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