I have a coordinate transformation:
\begin{equation} \mathbf{r} \rightarrow R \cdot \mathbf{r} \tag{1}\label{eq:transformation} \end{equation}
where $R$ is a constant orthogonal matrix. I have been looking through working and can not follow it. I was wondering if some light could be shed. It is stipulated that if \eqref{eq:transformation} then:
\begin{equation} \dot{\mathbf{r}} \rightarrow R \cdot \dot{\mathbf{r}} \tag{2}\label{eq:transformeddot} \end{equation}
is true. Is this always true only is $R$ is orthogonal? Then it states if \eqref{eq:transformation} then \eqref{eq:transformeddot} so that:
\begin{equation} \mathbf{r}^2 \rightarrow \mathbf{r} \cdot R \cdot R^T \cdot \mathbf{r} = \mathbf{r}^2 \tag{3}\label{eq:1} \end{equation}
implying that $|\mathbf{r}| = |\mathbf{r}|$
\begin{equation} \dot{\mathbf{r}}^2 \rightarrow \dot{\mathbf{r}} \cdot R \cdot R^T \cdot \dot{\mathbf{r}} = \dot{\mathbf{r}}^2 \tag{4}\label{eq:2} \end{equation}
I am unsure how \eqref{eq:transformeddot} and \eqref{eq:1} are found out. This is all the working out and I am trying to wrap my head around it as opposed to just accepting it. How do I arrive at those equations with more comprehensive mathematics?
Edit:
My thought process of obtaining $\mathbf{r}$ follows from \eqref{eq:transformation}:
$\textbf{r}^2 = \textbf{r} \cdot \textbf{r} \rightarrow \textbf{R} \cdot \textbf{r} \cdot \textbf{R} \cdot \textbf{r}$ and I am unsure how I simplify this.
Remember
If anyone has the same conundrum remember: $\textbf{r} \cdot \textbf{r} = \textbf{r}^T \textbf{r}$
First, some hints:
Use these identities in your derivations to get the resulting equations.
To elaborate on the second point, recall a dot product can be represented as $\bf{r} \cdot \bf{r} = \bf{r}^T \bf{r}$, thus after the transformation, the length of the vector is $(R \bf{r})^T (R \bf{r})=\bf{r}^T R^T R \bf{r}$, using the orthogonality property this is nothing but $\bf{r}^T\bf{r}$ which means the vector's magnitude is unchanged.