I have three points $a, b, c \in \mathbb{R^3}$. I want to get get a orthonormal basis for a unique plane which contains those three points. One of the ways to do it: from the three points get the scalar equation of a plane, chose arbitrary two vectors on the plane, G-S it. However, since I start already with points instead of plane equation, maybe it is possible to bypass the explicit construction of plane equation?
What I have tried is to chose two vectors $$\vec v_1 = b-a, \qquad \vec v_2 = c-b$$ and apply G-S to them. But the results I get seems to be incorrect. I think the plane I get will always go through point $(0, 0, 0)$, which is obviously incorrect result for arbitrary points $a, b, c$.
What am I missing here?
P.S. I need all of this to be able to transform a vector in $\mathbb{R^3}$ defined by three coordinates and belonging to the constructed plane, to a vector on the plane, defined by two coordinates, perform some procedures with it, which lets say, will result in another vector on the plane, then transform that vector back to three-coordinates definition.