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Equations of a Plane/Hyperplane

  1. Sep 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Given 3 points in 3-dimensional space, Find the equation of a plane containing those 3 points. How would you generalize this to n points in n-dimensional space?

    2. Relevant equations

    Equation of a Plane: ax + by + cz + d = 0 (1)
    Equation of a Hyperplane: [tex]a_1 x_1 + ... + a_n x_n + d=0[/tex] (2)

    3. The attempt at a solution

    For the 3-D case, I simply substituted the 3 points into the equation. For example, p1 = (x1, y1, z1), p2 = (x2, y2, z2), p3=(x3, y3, z3).

    ax1 + by1 + cz1 = -d
    ax2 + by2 + cz2 = -d
    ax3 + by3 + cz3 = -d

    ax1 + by1 + cz1 = ax2 + by2 + cz2
    => a(x1-x2) + b(y1-y2) + c(z1-z2) = 0

    Similarly,

    a(x1-x3) + b(y1-y3) + c(z1-z3) = 0

    and

    a(x2-x3) + b(y2-y3) + c(z2-z3) = 0

    Which is 3 equations with 3 unknowns that can be solved.

    Where I'm stumped is how I would generalize this to the n-dimensional case. I have a feeling that maybe I'm doing this the 'dumb' way and there's a far more elegant solution (perhaps involving matrices?) for solving the 3-D case that will extend more easily to the n-dimensional case.

    Any help would be appreciated!
     
  2. jcsd
  3. Sep 2, 2009 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi AngelofMusic! :smile:

    Hint: find the normal (and don't use coordinates, use whole vectors) …

    try the 3D case first: for three vectors a b and c, what can you say about the normal? :wink:
     
  4. Sep 2, 2009 #3
    The plane is the determinant:

    [tex]\begin{vmatrix}
    x-x_1 & y-y_1 & z-z_1\\
    x_2-x_1 & y_2-y_1 & z_2-z_1 \\
    x_3-x_1 & y_3-x_1 & z_3-z_1
    \end{vmatrix}=0[/tex]

    Where [itex]M_1(x_1,y_1), M_2(x_2,y_2), M_3(x_3,y_3)[/itex].
     
  5. Sep 2, 2009 #4
    One such normal would be n = (a-b)x(b-c), right? So for the n-dimensional case, would I just repeatedly take cross products of the vectors? The wikipedia page on surface normals has a neat solution where [tex]n=(AA^T + bb^T)^{-1} b[/tex], but they don't show how that is derived.

    Thanks a lot for the help so far! I can deal with 3-D cases relatively well, but my mind just isn't wrapping around the n-dimensional algebra very well at the moment.
     
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