When learning the position vector $\:\mathbf r$, I was told it was not a real vector because it is tied to a particular coordinate system. Now I have learned that the derivative of any vector $\:\mathbf C\:$ in a rotating frame and the inertial frame can be related via (equation \ref{9.8} in $^{\prime\prime}$An Introduction to Mechanics$^{\prime\prime}$ by D.Kleppner-R.Kolenkow, CUP 2nd Edition 2014), \begin{equation} \left(\frac{\mathrm d\mathbf C}{\mathrm d\, t}\right)_{in}\!\!\!\!\!=\left(\frac{\mathrm d\mathbf C}{\mathrm d\,t}\right)_{rot}\!\!\!\!\!+\boldsymbol{\Omega\times} \mathbf C \tag{9.8}\label{9.8} \end{equation} The author claims that $\:\mathbf C\:$ can be replaced by $\:\mathbf r$, then the velocity in a rotating frame can be related to the inertial velocity via \begin{equation} \mathbf v_{in}=\mathbf v_{rot}+\boldsymbol{\Omega\times} \mathbf r \end{equation} where $\:\mathbf v_{in}\:$ denotes the velocity in the inertial frame, $\:\mathbf v_{rot}\:$ denotes the velocity in the rotation frame. $\:\boldsymbol \Omega\:$ denotes the angular velocity of the rotating frame.
What confuses me is how we can replace $\:\mathbf C\:$ with $\:\mathbf r\:$ if $\:\mathbf r\:$ is not a real vector. Even if we can, I wonder to which coordinate system this $\:\mathbf r\:$ is tied, and where the origin is.
