Three Dimensional Geometry

Find the angle between the planes - Wired Faculty

Question

Find the angle between the planes
$straight r with rightwards arrow on top. space open parentheses 3 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space 0 space space and space straight r with rightwards arrow on top. space space left parenthesis 2 straight i with hat on top space minus space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis space equals space 0$

Solution

The equations of the planes are

straight r with rightwards arrow on top. space open parentheses 3 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space space 5 space straight k with hat on top close parentheses space equals space 0 space space space and space space straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis space equals space 0

therefore space space space space space stack straight n subscript 1 with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top comma space space space space stack straight n subscript 2 with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space minus space 2 space straight k with hat on top therefore space space space space open vertical bar stack straight n subscript 1 with rightwards arrow on top close vertical bar space equals space square root of 9 plus 16 plus 25 end root space equals space square root of 50 space equals space 5 square root of 2 space space space space space space space space open vertical bar stack straight n subscript 2 with rightwards arrow on top close vertical bar space equals space square root of 4 plus 1 plus 4 end root space space equals space square root of 9 space equals space 3

stack straight n subscript 1 with rightwards arrow on top. space stack straight n subscript 2 with rightwards arrow on top space equals space left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis negative 4 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 5 right parenthesis thin space left parenthesis negative 2 right parenthesis space equals space 6 plus 4 minus 10 space equals space 0

Let θ be the angle between the planes

therefore space space space space space space space space space cos space straight theta space equals space fraction numerator stack straight n subscript 1 with rightwards arrow on top. space stack straight n subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight n subscript 1 with rightwards arrow on top close vertical bar space open vertical bar stack straight n subscript 2 with rightwards arrow on top close vertical bar end fraction space equals fraction numerator 0 over denominator 5 square root of 2 cross times 3 end fraction space equals space 0 therefore space space space space space space space space space space straight theta space equals space 90 degree.