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Three Dimensional Geometry

Question

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

Solution

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

therefore space space straight l space equals cos space 45 degree space equals fraction numerator 1 over denominator square root of 2 end fraction comma space straight m space equals space cos space 45 degree space equals space fraction numerator 1 over denominator square root of 2 end fraction comma space straight n space equals space cos space 60 degree space equals space 1 half

These values of l, m, n do not satisfy the selection l² + m² + n² = 1 as

open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared plus space open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared plus space open parentheses 1 half close parentheses squared space equals space 1 half plus 1 half plus 1 fourth space equals 5 over 4 not equal to 1

∴ given angles cannot be the direction angles of a line.

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