Three Dimensional Geometry

Find the angle between the lines - Wired Faculty

Question

Find the angle between the lines
$straight x over 1 space equals fraction numerator negative straight y over denominator 0 end fraction space equals space fraction numerator straight z over denominator negative 1 end fraction space space and space space straight x over 3 space equals space straight y over 4 space equals straight z over 5$

Solution

The given lines are

straight x over 1 space equals space straight y over 0 space equals space fraction numerator straight z over denominator negative 1 end fraction space space space and space space straight x over 3 space equals space straight y over 4 space equals space straight z over 5

Let

stack straight b subscript 1 with rightwards arrow on top comma space stack straight b subscript 2 with rightwards arrow on top

be vectors parallel to these lines

therefore space space space space space stack straight b subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus space straight k with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top

Let θ be the angle between given lines.
∴ θ is angle between

stack straight b subscript 1 with rightwards arrow on top space space and space space stack straight b subscript 2 with rightwards arrow on top

therefore space space space space space space space cos space straight theta space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top. space stack straight b subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight b subscript 1 with rightwards arrow on top close vertical bar space open vertical bar stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator left parenthesis 1 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 1 plus 0 plus 1 end root space square root of 9 plus 16 plus 25 end root end fraction space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 3 minus 5 over denominator square root of 2 space square root of 50 end fraction space equals space fraction numerator negative 2 over denominator square root of 2 cross times 5 square root of 2 end fraction space equals space minus 1 fifth

∴ acute angle θ is given by

cos space straight theta space equals space 1 fifth

therefore space space space space space straight theta space equals cos to the power of negative 1 end exponent open parentheses 1 fifth close parentheses