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Math Help - Plane, Sphere

  1. #1
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    Plane, Sphere

    If you know the plane 2x+2y-z-18=0, you know that the normal vector to this plane is 2,2,-1. If you seek know direction vectors of the plane, can you say that (2,2,-1)(x,y,z)=0
    -->
    2x+2y-z=0 and then invent some points?
    Are therefore (1,-1,0),(-1,1,0),(1,1,4) etc. all direction vectors of the plane or am i making some kind of mistake here?
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  2. #2
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    Quote Originally Posted by Schdero View Post
    If you know the plane 2x+2y-z-18=0, you know that the normal vector to this plane is 2,2,-1. If you seek know direction vectors of the plane, can you say that (2,2,-1)(x,y,z)=0
    --> 2x+2y-z=0 and then invent some points?
    Are therefore (1,-1,0),(-1,1,0),(1,1,4) etc. all direction vectors of the plane or am i making some kind of mistake here?
    The is no such concept as direction vectors of the plane as far as I know.
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  3. #3
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    There is. Maybe I give it a wrong name, but if you look at the parametric quation of a plane you of one point + t(directionvector1) + u(directionvector2). These two directionvectors, if that's there name, are normal to the normal vector of the plane, which is in my case 2,2,-1
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  4. #4
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    Quote Originally Posted by Schdero View Post
    If you know the plane 2x+2y-z-18=0, you know that the normal vector to this plane is 2,2,-1. If you seek know direction vectors of the plane, can you say that (2,2,-1)(x,y,z)=0
    -->
    2x+2y-z=0 and then invent some points?
    Are therefore (1,-1,0),(-1,1,0),(1,1,4) etc. all direction vectors of the plane or am i making some kind of mistake here?
    1. In German the vectors you are looking for are called "span vectors" (literally translated).

    2. Since P(5,5,2) is a point in the plane the equation of the plane could be:

    \left(\begin{array}{c}x \\y \\ z \end{array}\right) = \left(\begin{array}{c}5 \\5 \\ 2\end{array} \right) + r \cdot \left(\begin{array}{c}1 \\-1 \\ 0\end{array} \right) + s \cdot \left(\begin{array}{c}1 \\1 \\ 4 \end{array} \right)

    which you easily can transform into the parametric equations of the plane.
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