Three Dimensional Geometry

Find the angle between the line and the plane - Wired Faculty

Question

Find the angle between the line
$straight r with rightwards arrow on top space equals space left parenthesis 2 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space straight lambda space left parenthesis 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis$ and the plane
$straight r with rightwards arrow on top. space left parenthesis 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis space equals space 4.$

Solution

The equation of line is

straight r with rightwards arrow on top space equals space left parenthesis 2 straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space straight lambda thin space left parenthesis 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis

therefore space space space space space space space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space minus space 3 straight j with hat on top space plus space 2 space straight k with hat on top

The equation of plane is

straight r with rightwards arrow on top. space open parentheses 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space 4 therefore space space space space space space space space space space space space space space space space space space space space space straight n with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top

Let θ be angle between the line and plane.

therefore space space space space space space sin space straight theta space equals space fraction numerator straight b with rightwards arrow on top. space straight n with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar. space open vertical bar straight n with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis negative 3 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 4 plus 9 plus 4 end root space square root of 9 plus 4 plus 25 end root end fraction

equals space fraction numerator 6 plus 6 plus 10 over denominator square root of 17 space square root of 38 end fraction space equals space fraction numerator 22 over denominator square root of 17 space square root of 38 end fraction

therefore space space space space straight theta space equals space sin to the power of negative 1 end exponent open parentheses fraction numerator 22 over denominator square root of 17 space square root of 38 end fraction close parentheses.