Assume a $X_{W}Y_{W}Z_{W}$ orthogonal coordinate system is transformed to an arbitrary orthogonal system $X_{C}Y_{C}Z_{C}$ by a 3D transformation matrix M, where M is represented by 4x4 matrix:
$$ \begin{bmatrix} a & b & c & t_x \\ d & e & f & t_y \\ g & h & i & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$O_{W}$ is the origin of $X_{W}Y_{W}Z_{W}$ and $O_{C}$ is the origin of $X_{C}Y_{C}Z_{C}$.
Given an arbitrary point P in the space. It's coordinates in $X_{W}Y_{W}Z_{W}$ is $(x_{w}, y_{w}, z_{w})$, while in $X_{C}Y_{C}Z_{C}$ is $(x_{c}, y_{c}, z_{c})$. The coordinates of $O_{C}$ in $X_{W}Y_{W}Z_{W}$ is $(x, y, z)$.
I wonder does the relation $\overrightarrow{O_WP} = \overrightarrow{O_WO_C} + \overrightarrow{O_CP}$ hold? It seems that $\overrightarrow{O_WO_C}$ and $\overrightarrow{O_CP}$ are in different coordinates and the coordinates unit may change.
If so, is it possible to add vectors like $\overrightarrow{O_WO_C}$ in $X_{W}Y_{W}Z_{W}$ with vectors like $\overrightarrow{O_CP}$ in $X_{C}Y_{C}Z_{C}$, or mathmatically, can we derive
$$ x_{w} = x_{c} + x \\ y_{w} = y_{c} + y \\ z_{w} = z_{c} + z $$
from $(x_{w}, y_{w}, z_{w}) = (x_{c}, y_{c}, z_{c}) + (x, y, z)$ ?
If matrix M is only a translation matrix, I think the relation holds. But I'm not sure about an arbitrary transformation.
