Rotate normal vector onto axis plane

Question

I have a set of data points in 3D space which apparently all fall onto a specific plane. I use PCA to compute the plane parameters. The 3rd component of PCA gives me the normal

Answer 1

How about:

Decompose the normal vector into a vector in the XY-plane and a Z vector. Then apply a rotation around the Z-axis to line up the XY vector with one of the axis. Then find the dot product of the the normal with the Z-axis, and rotate along which ever of X,Y you lined up with.

The idea is to line the normal vector up with Z, and by doing that your plane is now the XY plane.