Last updated at Dec. 8, 2016 by
Transcript
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 = 𝒂𝟏 𝒂𝟐 + 𝒃𝟏 𝒃𝟐 + 𝒄𝟏 𝒄𝟐 𝒂𝟏𝟐 + 𝒃𝟏𝟐 + 𝒄𝟏𝟐 𝒂𝟏𝟐 + 𝒃𝟏𝟐 + 𝒄𝟏𝟐 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) 𝑎2 + 𝑏2 + 𝑐2 𝑏 − 𝑐2 + 𝐶 − 𝑎2 + 𝑎 − 𝑏2 = 𝑎𝑏 − 𝑎𝑐 + 𝑏𝑐 − 𝑎𝑏 + 𝑐𝑎 − 𝑏𝑐 𝑎2 + 𝑏2 + 𝑐2 𝑏2 + 𝑐2 − 2𝑏𝑐 + 𝑐2 + 𝑎2 − 2𝑐𝑎 + 𝑎2 + 𝑏2 − 2𝑎𝑏 = 0 ∴ cos θ = 0 So, θ = 90° Therefore, angle between the given pair of lines is 90°
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