Three Dimensional Geometry

Find the angle between pair of lines: and - Wired Faculty

Question

Find the angle between pair of lines:
$fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 5 end fraction space equals space fraction numerator straight z plus 3 over denominator 4 end fraction$

and $fraction numerator straight x plus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 2 end fraction$

Solution

The equation of lines are

fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 5 end fraction space equals space fraction numerator straight z plus 3 over denominator 4 end fraction

and

fraction numerator straight x plus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 2 end fraction

Direction-ratios of the two lines are 3, 5, 4 and 1,1, 2.
Let θ be the angle between the lines.

therefore space space space cos space straight theta space equals space fraction numerator left parenthesis 3 right parenthesis left parenthesis 1 right parenthesis space plus space left parenthesis 5 right parenthesis left parenthesis 1 right parenthesis space plus space left parenthesis 4 right parenthesis left parenthesis 2 right parenthesis over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis 5 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared end root end fraction space space space space space space space space space space space space space space space space space equals space fraction numerator 3 plus 5 plus 8 over denominator square root of 50 square root of 6 end fraction space equals space fraction numerator 16 over denominator 5 square root of 2 space square root of 6 end fraction space equals space fraction numerator 16 over denominator 5 square root of 2. space square root of 2 space square root of 3 end fraction space equals space fraction numerator 8 over denominator 5 square root of 3 end fraction space equals space fraction numerator 8 square root of 3 over denominator 15 end fraction therefore space space space space straight theta space equals space cos space to the power of negative 1 end exponent space open parentheses fraction numerator 8 square root of 3 over denominator 15 end fraction close parentheses

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Three Dimensional Geometry

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