Finding an angle between rotated axis

[Aschie4589]

As from the title, I'm trying to find the angle between the y axis of two systems, with the same centre and rotated in respect to each other.
I have a vector that is measured in both the systems, so I have two sets of coordinates.
What I have tried to do is, consider the two sets of coordinates as if they were two different vectors and then, using scalar and vectorial product, find the angle and the rotational axis (so the transformation that turns one into the other), then apply Rodrigues' formula to transform the y axis using the same rule. After having the unitary vector for the y axis, I use again the scalar product (dot product (?)) to find the angle
Problem: I have to use this data in a wider experiment and, as far as I can tell, the angle that I find is wrong. I don't know why that is, I don't really know if I got the procedure right and the calculations wrong or the other way around...

 

[stapel]

I'm trying to find the angle between the y axis of two systems, with the same centre and rotated in respect to each other.

I have a vector that is measured in both the systems, so I have two sets of coordinates.

What I have tried to do is consider the two sets of coordinates as if they were two different vectors and then, using scalar and vectorial product, find the angle and the rotational axis (so the transformation that turns one into the other), then apply Rodrigues' formula to transform the y axis using the same rule. After having the unitary vector for the y axis, I use again the scalar product (dot product (?)) to find the angle.

Problem: I have to use this data in a wider experiment and, as far as I can tell, the angle that I find is wrong. I don't know why that is, I don't really know if I got the procedure right and the calculations wrong or the other way around...

Please reply with a clear listing of your steps and your results, along with the "correct" values. Thank you!

 

[Paul Belino]

Sounds like you are confused. Why don't you just do it from first principles? Forget vectors and stuff and just concentrate on the trigonometry of it. Construction two right hand triangles of sides x,y in one systems and sides x' and y' in the other and angle theta between them. It just boils down to finding an expression for x' and y' in terms of x,y,and theta. Thne go back and look at the vector method of doing it. The maths is the same its just disguised in the vector notation.