Three Dimensional Geometry

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

Question

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

Solution

The equation of line is

straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space minus space straight k with hat on top close parentheses space space plus space straight lambda space open parentheses straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

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

The equation of plane is

straight r with rightwards arrow on top. space space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top right parenthesis space equals space 4

therefore space space space space straight n 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 plus space straight k with hat on top

Let θ be the angle between the line and the plane

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 open parentheses straight i with hat on top space minus space straight j with hat on top plus space straight k with hat on top close parentheses space open parentheses 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses over denominator square root of 1 plus 1 plus 1 end root space square root of 4 plus 1 plus 1 end root end fraction space equals space fraction numerator 2 plus 1 plus 1 over denominator square root of 3 space square root of 6 end fraction space equals space fraction numerator 4 over denominator 3 square root of 2 end fraction therefore space space sin space straight theta space equals space fraction numerator 2 square root of 2 over denominator 3 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 2 square root of 2 over denominator 3 end fraction close parentheses.