The 2D part is due to the touch screen coordinates. However, my rotation matrix is for manipulating 3D vertices, where the angle theta is required. It's a touchscreen FPS app after all.

I need to find the angle from the X axis to the red line. I don't understand how (cos(angle), sin(angle) = normalized(P2-P1) when the angle is arbitrary in the range 0 degrees to 90 degrees. Unless you mean that for given P1 and P2, by doing arccosine and arcsine on the normalized X and Y of the vector, I can obtain the angle?

You keep saying that you need to find the angle, but knowing (cos(angle), sin(angle)) is usually enough. You get that simply from normalizing P2-P1, as you say. If you really need the angle, you can use atan2. The code looks something like this:

Vector2D diff = P2-P1;
float angle = atan2(diff.y,diff.x);

My mind cracked into place. Now, it is obvious to me.

Thank you for all of your time. Problem solved.