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Surface formed by set of points perpendicular vectors

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I'm reviewing for my Calc III course and I came across a problem that's stumped me and I can't easily find any example of on the web.

Given point A (0, 2, 2) and B (2, 2, 2), find the set of all points P such that vector AP is perpendicular to vector BP.

My first reaction was thinking this was easy, and I just set up an equation with one unknown P, where I let the dot product = 0 (as the vectors should be perpendicular). This gives me a quadratic equation and thus two values for P. However, I'm afraid this is wrong as my options for the surface (I also have to write the surface's equation) are plane/line/sphere/cone/parabaloid/hyperboloid.


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For ease of computation I'm going to translate A--> A'(0,0,0), B --> B' (2,0,0) (a translation of (0, -2, -2)). Translation does not distort geometry.
A'P = (x, y, z), B'P = (x-2, y, z). set dot product = 0
x^2 - 2x + y^2 + z^2 = 0. 
(x - 1)^2 + y^2 + z^2 = 1
This is a sphere, where AB forms the diameter of the sphere. 

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