The general form equation of a plane in $R^3$ is:
$$ax + by + cz = d$$
I am given 3 points and asked to derive a general form equation for a plane passing through those three points, say, A, B and C.
The approach I take is to:
Choose one of those three points, say, A
Find the vector from A to B: $\vec {v_1} = \vec b - \vec a$
Find the vector from A to C: $\vec {v_2} = \vec c - \vec a$
Find the normal vector, n, to the plane by finding the cross product of said vectors $$\vec n = \vec v_1 \times \vec v_2$$
Use the normal form equation for a plane in $R^3$, i.e. $\vec n \bullet \vec x = \vec n \bullet \vec p$, to substitute in the normal vector, n, calculated above and the point from which it was calculated, in this case, A.
If the answer that I get is correct then I should expect substituting any point A B or C into the equation $$ax + by + cz = d$$ ... to hold. However it doesn't. The equation only holds for one point, A, and not B and C.
What am I doing incorrectly to derive the equation of a plane passing through point A, B and C?