Three Dimensional Geometry

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

Question

Find the angle between the following pair of lines:
$fraction numerator straight x minus 2 over denominator 2 end fraction space equals space fraction numerator straight y minus 1 over denominator 5 end fraction space equals fraction numerator straight z plus 3 over denominator negative 3 end fraction space space and space space fraction numerator straight x plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight y minus 4 over denominator 8 end fraction space equals space fraction numerator straight z minus 5 over denominator 4 end fraction$

Solution

The equations of lines are

fraction numerator straight x minus 2 over denominator 2 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 negative 3 end fraction

and

fraction numerator straight x plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight y minus 4 over denominator 8 end fraction space equals space fraction numerator straight z minus 5 over denominator 4 end fraction

Direction ratios of two lines are 2, 5, -3 and – 1, 8, 4.
Let θ be the angle between the lines.

therefore space space space cos space straight theta space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 5 right parenthesis thin space left parenthesis 8 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 4 right parenthesis over denominator square root of 4 plus 25 plus 9 end root space square root of 1 plus 64 plus 16 end root end fraction space equals space fraction numerator negative 2 plus 40 minus 12 over denominator square root of 38 space square root of 81 end fraction space space space space space space space space space space space space space space space space space equals fraction numerator 26 over denominator 9 square root of 38 end fraction therefore space space space space space space space space space space straight theta space equals space cos to the power of negative 1 end exponent space open parentheses fraction numerator 26 over denominator 9 square root of 38 end fraction close parentheses space