Find the acute angle between the lines
(x - 4)/3 = (y + 3)/4 = (z + 1)/5 and (x - 1)/4 = (y + 1)/(-3) = (z + 10)/5
Last updated at Oct. 18, 2019 by Teachoo
Transcript
Question 25 Find the acute angle between the lines (๐ฅ โ 4)/3 = (๐ฆ + 3)/4 = (๐ง + 1)/5 and (๐ฅ โ 1)/4 = (๐ฆ + 1)/(โ3) = (๐ง + 10)/5 Angle between the pair of lines (๐ฅ โ ๐ฅ_1)/๐_1 = (๐ฆ โ ๐ฆ_1)/๐_1 = (๐ง โ ๐ง_1)/๐_1 and (๐ฅ โ ๐ฅ_2)/๐_2 = (๐ฆ โ ๐ฆ_2)/๐_2 = (๐ง โ ๐ง_2)/๐_2 is given by cos ฮธ = |(๐_1 ๐_2 + ๐_1 ๐_2 + ๐_1 ๐_2)/(โ(ใ๐_1ใ^2 + ใ๐_1ใ^2 + ใ๐_1ใ^2 ) โ(ใ๐_2ใ^2 + ใ๐_2ใ^2 + ใ๐_2ใ^2 ))| (๐ โ ๐)/๐ = (๐ + ๐)/๐ = (๐ + ๐)/๐ Comparing with (๐ฅ โ ๐ฅ_1)/๐_1 = (๐ฆ โ ๐ฆ_1)/๐_1 = (๐ง โ ๐ง_1)/๐_1 ๐1 = 3, b1 = 4, c1 = 4 (๐ โ ๐)/๐ = (๐ + ๐)/(โ๐) = (๐ + ๐๐)/๐ Comparing with (๐ฅ โ ๐ฅ_2)/๐_2 = (๐ฆ โ ๐ฆ_2)/๐_2 = (๐ง โ ๐ง_2)/๐_2 ๐2 = 4, ๐2 = โ3, ๐2 = 5 Now, cos ฮธ = |(๐_1 ๐_2 + ๐_1 ๐_2 + ๐_1 ๐_2)/(โ(ใ๐_1ใ^2 + ใ๐_1ใ^2 + ใ๐_1ใ^2 ) โ(ใ๐_2ใ^2 + ใ๐_2ใ^2 + ใ๐_2ใ^2 ))| = |(3 ร 4 + 4 ร (โ3) + 5 ร 5)/(โ(3^2 + 4^2 + 5^2 ) โ(4^2 +(โ3)^2 + 5^2 ))| = |(12 โ 12 + 25)/(โ(9 + 16 + 25) โ(16 + 9 + 25))| = |25/(โ50 โ50)| = |25/50| = |1/2| = 1/2 So, cos ฮธ = 1/2 โด ฮธ = 60ยฐ = ๐ /๐ Therefore, required angle is ๐ /๐ Note: Please write angle in radians and not degree
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