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Homework Help: (Multivariable Calc) Given four points, find a sphere . . . help me.

  1. Aug 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the radius and center for the sphere determined by the four points
    (3,2,-1), (5,-2,1), (-3,1,2), (0,2,4).
    Give the exact location of the center but approximate the radius to the nearest 0.1.

    2. Relevant equations
    Equation of a sphere: x2 + y2 + z2 + Gx +Hy +Iz +J = 0
    or C(h,k,l) (x-h)2 + (y-k)2 + (z-l)2 - r2 = 0

    There's also a 5x5 matrix with the determinant equal to zero, but I have no idea how to use it (seen at line 29 here: http://mathworld.wolfram.com/Sphere.html). I believe this is beyond 12.1 in multivariable calculus.

    3. The attempt at a solution
    I tried to use the determinant formula, but I got lost as I don't even know how to use it.

    Is there any other way?

    EDIT: Solved. Thank you.
    Last edited: Aug 30, 2009
  2. jcsd
  3. Aug 30, 2009 #2


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    (x-h)2 + (y-k)2 + (z-l)2 - r2 = 0

    Using this, I'd just substitute the 4 coordinates to get me 4 equations with 4 unknowns in them. Then just use row reduction to get h,k,l and r
  4. Aug 30, 2009 #3
    So I can substitute, then treat the different lines as a system of equations?
  5. Aug 30, 2009 #4


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    well your variables would be h2,k2,l2,r2 and not h,k,l,r like I previously said.
  6. Aug 30, 2009 #5
    Alright I will try that. Thank you very much.
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