This is a purely math question, I believe.

I have a camera object in an arbitrary coordinate system. I have the direction vector of the camera, and I have a vector that points in the direction of north along the surface of the sphere.

I wish to calculate the angle of the camera in regards to the north vector. Is there a simple way to calculate the final sign of the angle? I'm aware than angle = acos(dir dot north), but when I implement it, the angle is clamped to [0,180] degrees. Is there a way to figure out whether the camera is painting eastward or not?