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Three Dimensional Geometry

Question

Can a directed line have direction angles 45°, 60°, 120°?

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Solution

Let l, m, n be the direction cosines of the line with direction angles 45°. 60°, 120°
∴ l = cos 45°

equals space fraction numerator 1 over denominator square root of 2 end fraction comma space space straight m space equals space cos space 60 degree space equals 1 half comma space space straight n space equals cos space 120 degree space equals space cos space left parenthesis 180 degree minus 60 degree right parenthesis space equals space minus cos space 60 degree space equals negative 1 half

therefore space space space straight l squared plus straight m squared plus straight n squared space equals space 1 half plus 1 fourth plus 1 fourth space equals space fraction numerator 2 plus 1 plus 1 over denominator 4 end fraction space equals 4 over 4 space equals 1

∴ a line can have the given angles as direction angles.

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