If (1,-1,2) and (2,1,-1) are the end points of a diameter of a sphere x^{2}+y^{2}+z^{2}+2ux+2vy+2wz-1=0, then what is u+v+w equal to?

Explanation

The centre of the sphere is (-u, -v, -w). This centre must be the midpoint of the given diameter endpoints (1,-1,2) and (2,1,-1). The midpoint is (\frac{1+2}{2}, \frac{-1+1}{2}, \frac{2-1}{2}) = (\frac{3}{2}, 0, \frac{1}{2}). Thus, -u = \frac{3}{2}, -v = 0, and -w = \frac{1}{2}, yielding u = -\frac{3}{2}, v = 0, w = -\frac{1}{2}. The sum u+v+w = -\frac{3}{2} + 0 - \frac{1}{2} = -2.

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