I am currently doing an online course in Udemy to clarify some things for a project I'm currently doing on solid rigid motion and simulation in University, although understanding some of the things regarding the use of rotation matrixes has confused for some time already.
Given a coordinate system $\{XYZ\}_I$, and a sequence of rotations in $Z$, $Y$, $X$ represented by the matrices $R_z$,$R_y$,$R_x$ respectively performed over the original axis and not the one after each rotations that rotate $\{XYZ\}_I$ to $\{XYZ\}_B$ , and given $u_B \in \mathbb{R}^3$, I do not understand completely the way to calculate $u_I$ is: \begin{equation} u_I = R_xR_yR_zu_B \label{eq1} \end{equation}
What I understand of Rotation matrices is that a rotation from a frame $\{XYZ\}_A$ to an rotated frame $\{XYZ\}_B$ can be described as a rotation matrix $R$ that maps a vector from the rotated frame $B$ to an original frame $A$ the following way:
$$u_A = R u_B$$
Following that intuition, by calculating the rotations as in the first equation it would imply that the $Rz$ maps the vector $u_B$ to an intermediate frame (let's say $B_z$), then $R_y$ maps from $B_z$ to another intermediate frame (let's say $B_y$), and finally, $R_x$ rotates from $B_y$ to the inertial and original body frame $I$.
According to the course, that is not like that because the rotations in this case are described as rotations from the inertial frame to each of the intermediate body frames. That would mean that $Rz$ maps from the first intermediate frame to the inertial frame (instead of maping form the final body frame to the first intermediate), and then, the next rotation would be mapping from the inertial frame to the inertial frame again (?) since the vector that was returned in the previous transformation is to the inertial frame, but that is a contradiction unless all the rotations left are the identity.
I know there is something I'm not getting, and I know there is also the possibility of post-multiplying instead of pre-multiplying, which I guess that implys doing the opposite rotaiton, although I'm still a bit lost and don't totally understand this elements.
Also, even if I can kind of see how it works if we are rotating from a given frame to another frame, and a sequence of rotations, how would we define a sequence of rotations if they are all respect to the world, inertial frame?
If $x $ is a vector in an initial coordinate system $B_i$ , the corresponding the world coordinates is the vector $y$ where
$ y = B_i \ x $
If you now apply $3$ consecutive rotations to vector $y$ about the world axes $z$, $y$, $x$ in that order, then the final image of $y$ is the vector $y'$ where
$y' = R_x\ R_y\ R_z\ y = R_x\ R_y\ R_z\ B_i\ x $
which means that the final coordinate system is
$B_f = R_x\ R_y\ R_z\ B_i $
So that the relation between a vector in frame $B_i$ and its corresponding coordinate vector $y'$ after applying the rotations, is
$ y' = B_f\ x $