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Misc 3 - Find angle between lines whose direction ratios are - Miscellaneous

Misc 3  - Chapter 11 Class 12 Three Dimensional Geometry - Part 2

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Misc 3 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 ∴ cos θ = 0 So, θ = 90° Therefore, angle between the given pair of lines is 90°

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.