Spin axis and carom angles

I think any explanation of 3D angular momentum for a sphere will include this. The actual axis of rotation of a ball can simply be decomposed into its components along the major axes. You just take the x, y, and z components of the spin. It is very, very simple. You take the projection of the physical axis onto the analysis axes. That is the length of the spin vector multiplied by the cosine of the angle between the spin vector and the axis.

I had missed how to get the first spin vector, know everything makes sense. Sometimes in 3c you play a very thick hit with full side so I just wanted to know how to manipulate the angle from the tangent line to a line that is either shorter or longer than it. Thank you!