Three Dimensional Geometry

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

Question

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

Solution

The equation of line is $straight r with rightwards arrow on top space equals space open parentheses straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses space plus space straight lambda space open parentheses 2 straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses$
$therefore space space space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus 3 space straight k with hat on top$
The equation of plane is $straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space 4$
$therefore space space space space space straight n with rightwards arrow on top space equals space space 2 straight i with hat on top plus straight j with hat on top space minus space straight k with hat on top$
Let θ be the angle between the line and plane $therefore space space sin space straight theta space equals space fraction numerator straight b with rightwards arrow on top. space straight n with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight n with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis negative 1 right parenthesis over denominator square root of 4 plus 1 plus 9 end root space square root of 4 plus 1 plus 1 end root end fraction space equals space fraction numerator 0 over denominator square root of 4 space square root of 6 end fraction space equals space 0$

$therefore space space space straight theta space equals space 0 degree$
∴ line is parallel to the plane.

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Three Dimensional Geometry

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