Equation for Line of Plane?

Okay so. I'm pretty sure I have it done.

I have the two points already, (x[SUB]1[/SUB], y[SUB]1[/SUB], z[SUB]1[/SUB]) and (x[SUB]2[/SUB], y[SUB]2[/SUB], z[SUB]2[/SUB]).

I know from plane Ax + By + Cz = D that there is a normal vector n = Ai + Bj + Ck, which gives the position (A, B, C).

Now what I have three points I set up a matrix

[x y z 1
x1 y1 z1 1
x2 y2 z2 1
A B C 1] equal to =0

From there I'll make matrix:

[x-x1 y-y1 z-z1
x2-x1 y2-y1 z2-z1
A-x1 B-y1 C-z1] = 0

And solve the determinate of that. Hehehe I'll run it by my teacher, class is at 5 PM today. Thanks for the help you math geniuses!