Definition

Consider two reference frames $\vec{\mathcal{F}}_{a},\vec{\mathcal{F}}_{b}$ found at the same origin: $$\vec{r}=\vec{\mathcal{F}}_{a}\mathbf{r}_{a}=\vec{\mathcal{F}}_{b}\mathbf{r}_{b}$$The vector components $\mathbf{r}_{a},\mathbf{r}_{b}$ are related via the dot product of the frames:$$\begin{align*} \vec{\mathcal{F}}_{b}\mathbf{r}_{b}&=\vec{\mathcal{F}}_{a}\mathbf{r}_{a}\\ \underbrace{ \vec{\mathcal{F}}_{b}\cdot\vec{\mathcal{F}}_{b}^{\top} }_{ \text{orthogonal} }\mathbf{r}_{b}&=\underbrace{ \vec{\mathcal{F}}_{b}\cdot \vec{\mathcal{F}}_{a}^{\top} }_{ =:\mathbf{C}_{ba} }\mathbf{r}_{a}\\ \mathbf{r}_{b}=\mathbf{C}_{ba}\mathbf{r}_{a} \end{align*}$$where $$\mathbf{C}_{ba}:=\vec{\mathcal{F}}_{b}\cdot \vec{\mathcal{F}}_{a}^{\top}=\begin{bmatrix}\vec{b}_{1} \\ \vec{b}_{2} \\ \vec{b}_{3}\end{bmatrix}\cdot \begin{bmatrix}\vec{a}_{1}&\vec{a}_{2}&\vec{a}_{3}\end{bmatrix}$$is the rotation matrix to frame $b$ from frame $a$.

Properties

1. A rotation preserves the length of the vector and the orientation of space
2. A rotation matrix is a representation of a rotation as a $3\times3$ orthonormal matrix.