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

1. Aug 30, 2009

### Vampire

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. Aug 30, 2009

### rock.freak667

(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

3. Aug 30, 2009

### Vampire

So I can substitute, then treat the different lines as a system of equations?

4. Aug 30, 2009

### rock.freak667

well your variables would be h2,k2,l2,r2 and not h,k,l,r like I previously said.

5. Aug 30, 2009

### Vampire

Alright I will try that. Thank you very much.