Please help me with this vector problem

Sep 2019
Hello, i'm working on an assignment where i have to find the rotation ThetaR around the z axis. I have two vectors lets call them vector a and b
Known values are: phi-a, theta- a and phi-b and theta-b but they are arbitary. As you can see in the picture i have to find the rotation around the z axis but i also have a tilt.
If i find the normal vector of these two vectors and compare this normal vector to the z axis i have my tilt as can be seen in the picture. But how do i now find theta R?
I know that x= sin (theta) x cos(phi) and y = sin (theta) x sin (phi) and z = cos(phi)
so the tilt can be calculated if you take the inner product of the two vectors with [ 0 0 1] which is de z- axis but what to do to find thetaR?


Jun 2019
Never been great at working in 3-D -- in fact, that's how I ended up on this forum in the first place -- but I think I can take a stab at it.

The cross product $\vec{a} \times \vec{b}$ will give you $\vec{N}$ to within a constant, right? Then if I'm understanding the problem correctly, you're looking for the angle of the projection of N in the x-y plane, yes? So, $\theta_R = tan^{-1}(y_N/x_N)$?

*Incidentally, if you want $theta_R$ to be quadrant specific, you need to decide on a direction for the normal vector. I.e., in the direction of a cross b or of b cross a. Then use the quadrant-specific arctangent.
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