Three Dimensional Geometry

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

Question

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

Solution

The equations of two lines are

straight x over 2 space equals space straight y over 2 space equals space straight z over 1 space space space and space fraction numerator straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y minus 2 over denominator 1 end fraction space equals space fraction numerator straight z minus 3 over denominator 8 end fraction

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

therefore space space space space space cos space straight theta space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 8 right parenthesis over denominator square root of 4 plus 4 plus 1 end root space square root of 16 plus 1 plus 64 end root end fraction space equals space fraction numerator 8 plus 2 plus 8 over denominator square root of 9 space square root of 81 end fraction space equals space fraction numerator 18 over denominator 3 cross times 9 end fraction equals space 2 over 3 therefore space space space space space space space straight theta space equals space cos to the power of negative 1 end exponent open parentheses 2 over 3 close parentheses