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Math Help - Vectors and planes

  1. #1
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    Vectors and planes

    Please help. A ray of light coming from the point (-1, 3, 2) is travelling in the directionof the vector 4i + j -2k and meets the plane x + 3y + 2z - 24= 0. Find the angle that the ray of light makes with the plane.
    Tks!
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  2. #2
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    Hello, pantera!

    A ray of light coming from the point (-1, 3, 2) is travelling in direction \langle 4, 1,-2\rangle
    and meets the plane x + 3y + 2z - 24\:=\: 0
    Find the angle that the ray of light makes with the plane.
    The angle between two vectors \vec{u}\text{ and }\vec{v} is given by: . \cos\theta \:=\:\frac{|\vec{u}\cdot \vec{v}|}{|\vec{u}||\vec{v}|}

    The ray has direction: . \vec{u} \:=\:\langle4,1,-2\rangle
    The plane has normal direction \vec{n} \:=\:\langle 1, 3, 2\rangle

    The angle \theta between the ray and the normal is:
    . . \cos\theta \:=\:\frac{\langle 4,1,-2\rangle\cdot\langle1,3,2\rangle}{\sqrt{4^2+1^2+(-2)^2}\sqrt{1^2+3^2+2^2}} \;=\;\frac{3}{\sqrt{294}} \:=\:0.174963553

    Hemce, \theta \;=\;79.92346289 \;\approx\;80^o


    Therefore, the angle between the ray and the plane is about: . 90^o - 80^o \;=\;\boxed{10^o}

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