Three Dimensional Geometry

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

Question

Find the angle between the line $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus 3 space straight j with hat on top space plus space 9 space straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top right parenthesis$ and the plane $straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 5$

Solution

The equation of line is

straight r with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 9 straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 4 straight k with hat on top right parenthesis

therefore space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top

The equation of plane is

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

therefore space space space straight n with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top

Let θ be the angle between the line and plane

therefore 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 1 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of 4 plus 9 plus 16 end root space square root of 1 plus 1 plus 1 end root end fraction therefore space space sin space straight theta space equals space fraction numerator 9 over denominator square root of 29 space square root of 3 end fraction space equals space fraction numerator 3 square root of 3 over denominator square root of 29 end fraction space space space space space space rightwards double arrow space space space space space straight theta space equals sin to the power of negative 1 end exponent open parentheses fraction numerator 3 square root of 3 over denominator square root of 29 end fraction close parentheses