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I am the moment learning about rotation matrices. It seems confusing how it could be that $$R_A^C=R_A^B⋅R_B^C$$$$R_A^C=R_A^BR_B^C$$ is the rotation from coordinate frame A to C C to A, and A,B,C are different coordinate frames.

$$R_A^C$$ must for a 2x2 matrix be defined as $$R_A^C= \left( \begin{matrix} xa⋅xb & xa⋅xb \\ ya⋅yb & ya⋅yb \end{matrix} \right)$$

$$x_a, y_a and x_b,y_b$$ are coordinates for points given in different coordinate frame. I don't see how, using this standard, the multiplication stated above will give the same matrix as for $$R_A^C$$. Some form for clarification would be helpful here.

formatted matrix as such

I am the moment learning about rotation matrices. It seems confusing how it could be that $$R_A^C=R_A^B⋅R_B^C$$ is the rotation from coordinate frame A to C C to A, and A,B,C are different coordinate frames.

So.. $$R_A^C$$ must for a 2x2 matrix be defined as $$[xa⋅xb ~~~ xa⋅xb ~~~~ ; ~~~~ ya⋅yb ~~~ ya⋅yb]$$ $$R_A^C= \left( \begin{matrix} xa⋅xb & xa⋅xb \\ ya⋅yb & ya⋅yb \end{matrix} \right)$$ I don't see how, using this standard, the multiplication stated above will give the same matrix as for $$R_A^C$$. Some form for clarification would be helpful here.

tried to clarify

I am the moment learning about rotation matrices. It seems confusing how it could be that $$R_A^C=R_A^B⋅R_B^C$$ is the rotation from coordinate frame A to C C to A, and A,B,C are different coordinate frames.

So.. $$R_A^C$$ must for a 2x2 matrix be defined as $$[xa⋅xb ~~~ xa⋅xb ~~~~ ; ~~~~ ya⋅yb ~~~ ya⋅yb]$$ I don't see how, using this standard, the multiplication stated above will give the same matrix as for RCA$$R_A^C$$. Some form for clarification would be helpful here.

tried to clarify