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Misc 1 Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b. Angle between the lines with direction ratios a1, b1, c1 and a2, b2, c2 is given by cos θ = |(𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 + 𝒄_𝟏 𝒄_𝟐)/(√(𝒂_𝟏^𝟐 + 𝒃_𝟏^𝟐 + 𝒄_𝟏^𝟐 ) √(𝒂_𝟏^𝟐 + 𝒃_𝟏^𝟐 + 𝒄_𝟏^𝟐 ))| Given, 𝑎1 = 𝑎, 𝑏1 = 𝑏, c1 = c and 𝑎2 = 𝑏 − 𝑐, 𝑏2 = 𝑐 − 𝑎, c2 = a – b So, cos θ = |(𝑎(𝑏 − 𝑐) + 𝑏(𝑐 − 𝑎) + 𝑐(𝑎 − 𝑏))/(√(𝑎^2 + 𝑏^2 + 𝑐^2 ) √((𝑏 − 𝑐)2 + (𝐶 − 𝑎)2 + (𝑎 − 𝑏)2))| = |(𝒂𝒃 − 𝒂𝒄 + 𝒃𝒄 − 𝒂𝒃 + 𝒄𝒂 − 𝒃𝒄)/(√(𝑎^2 + 𝑏^2 + 𝑐^2 ) √(𝑏^2 + 𝑐2 − 2𝑏𝑐 + 𝑐^2 + 𝑎^2 − 2𝑐𝑎 + 𝑎^2 + 𝑏^2 − 2𝑎𝑏 ))| = |𝟎/(√(𝑎^2 + 𝑏^2 + 𝑐^2 ) √(𝑏^2 + 𝑐2 − 2𝑏𝑐 + 𝑐^2 + 𝑎^2 − 2𝑐𝑎 + 𝑎^2 + 𝑏^2 − 2𝑎𝑏 ))| = 0 Since cos θ = 0 So, θ = 90° Therefore, angle between the given pair of lines is 90°

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo