Three Dimensional Geometry

Find the angle between each of the following pairs of lines: - Wired Faculty

Question

Find the angle between each of the following pairs of lines:
$fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 2 end fraction comma space space straight z space equals space 2 comma space space space fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator 2 straight y plus 3 over denominator 3 end fraction space equals space fraction numerator straight z plus 5 over denominator 2 end fraction$

Solution

The equations of the two lines are

fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 0 end fraction comma space space space fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y plus begin display style 3 over 2 end style over denominator begin display style 3 over 2 end style end fraction space equals space fraction numerator straight z plus 5 over denominator 2 end fraction

Direction-ratios of two lines are

3 comma space minus 2 comma space 0 space space and space 1 comma space space 3 over 2 comma space 2 space space space space or space space space 2 comma space 3 comma space 4.

Let θ be the angle between the lines.

therefore

cos space straight theta space equals space fraction numerator left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis negative 2 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis 4 right parenthesis over denominator square root of 9 plus 4 plus 0 end root space square root of 4 plus 9 plus 16 end root end fraction space space equals space 0

rightwards double arrow

straight theta space equals space 90 degree