Three Dimensional Geometry

Question

Find the angle between the planes whose vector equations are
$straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space 2 straight j with hat on top space minus space 3 straight k with hat on top close parentheses space equals space 5 space space space and space straight r with rightwards arrow on top. space open parentheses 3 straight i with hat on top space minus space 3 straight j with hat on top space plus space 5 straight k with hat on top close parentheses space equals space 3.$

Solution

The equations of two planes are

straight r with rightwards arrow on top. space open parentheses 2 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top close parentheses space equals space 5

and

straight r with rightwards arrow on top. space space left parenthesis 3 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis space equals space 3

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

open vertical bar stack straight n subscript 1 with rightwards arrow on top close vertical bar space equals space square root of 4 plus 4 plus 9 end root space equals space square root of 17 comma 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 9 plus 9 plus 25 end root space equals space square root of 43

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 2 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis 5 right parenthesis space equals space 6 minus 6 minus 15 space equals space minus 15

Let θ be angle between the planes

therefore 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 space fraction numerator negative 15 over denominator square root of 17 space square root of 43 end fraction space equals space minus fraction numerator 15 over denominator square root of 731 end fraction

∴ acute angle θ is given by

cos space straight theta space equals space fraction numerator 15 over denominator square root of 731 end fraction space space space space space space space space rightwards double arrow space space space straight theta space equals space cos space to the power of negative 1 end exponent space open parentheses fraction numerator 15 over denominator square root of 731 end fraction close parentheses.