I had been under the impression that two points define a line, and three a plane.

So assuming we have three (non-collinear) points: A, B and C, with position vectors

and

, respectively. Then, by considering a general point on the plane, P, with position vector

, and making a parallelogram, it is possible to show that

So, in my eyes, this method seems to uniquely define the plane.

However, if we say that

, and

and

, then considering that the general form for a plane is

, we have but three equations:

,

and

,

and four variables. Therefore, there must exist multiple solutions. So, it seems to me, that this method does not define a plane uniquely - which leaves me a little confused, as I cannot imagine how three points could not define a plane uniquely.

Any suggestions much appreciated.