Misc 5 - Chapter 11 Class 12 Three Dimensional Geometry

Transcript

Misc 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 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 𝑎1, 𝑏1, c1 and 𝑎﷮2﷯ , 𝑏﷮2﷯, 𝑐﷮2﷯ is given by cos θ = 𝒂﷮𝟏﷯ 𝒂﷮𝟐﷯ + 𝒃﷮𝟏﷯ 𝒃﷮𝟐﷯ + 𝒄𝟏𝒄𝟐﷮ ﷮ 𝒂﷮𝟏﷯﷮𝟐﷯ + 𝒃﷮𝟏﷯﷮𝟐﷯+ 𝒄﷮𝟏﷯﷮𝟐﷯﷯ ﷮ 𝒂﷮𝟐﷯﷮𝟐﷯ + 𝒃﷮𝟐﷯﷮𝟐﷯+ 𝒄﷮𝟐﷯﷮𝟐﷯﷯﷯﷯ A line passing through A ( 𝑥﷮1﷯, 𝑦﷮1﷯, 𝑧﷮1﷯) and B ( 𝑥﷮2﷯, 𝑦﷮2﷯, 𝑧﷮2﷯) has direction ratios ( 𝑥﷮2﷯ − 𝑥﷮1﷯), ( 𝑦﷮2﷯ − 𝑦﷮1﷯), ( 𝑧﷮2﷯ − 𝑧﷮1﷯) = 68﷮ ﷮34﷯ ﷮136﷯﷯﷯ = 68﷮ ﷮34﷯ ﷮4 × 34﷯﷯﷯ = 68﷮ ﷮34﷯ × ﷮4﷯ × ﷮34﷯﷯﷯ = 68﷮ ﷮34﷯ × ﷮34﷯× ﷮4﷯﷯﷯ = 68﷮34 × 2 ﷯﷯ = 68﷮68﷯﷯ = 1 ∴ cos θ = 1 So, θ = 0° Therefore, angle between AB and CD is 0° .