3D Numanclature

We have a reference frame denoted as

$$\overrightarrow{{\mathcal{F}}_i}=\left[\begin{array}{c}{\vec{1}}_{1,i}\\ {\vec{1}}_{2,i}\\ {\vec{1}}_{3,i}\end{array}\right]$$

A vector in that reference frame is denoted as

$$\overrightarrow{r_i}=\left[\begin{array}{ccc}{r}_{1,i}& r_{2,i}& r_{3,i}\end{array}\right]\overrightarrow{{\mathcal{F}}_i}\overrightarrow{r_i}={\mathbf{r}}^T\overrightarrow{{\mathcal{F}}_i}={\overrightarrow{{\mathcal{F}}_i}}^T\mathbf{r}$$

Axes 1, 2, 3 are arbitrarily names, but they are orthogonal.

Cross Product

The cross product of two vectors in the same reference frame is given as

$$\vec{r}\times \vec{s}=[r_1{r}_2{r}_3]\left[\begin{array}{ccc}{\vec{1}}_1\times {\vec{1}}_1& {\vec{1}}_1\times {\vec{1}}_2& {\vec{1}}_1\times {\vec{1}}_3\\ {\vec{1}}_2\times {\vec{1}}_1& {\vec{1}}_2\times {\vec{1}}_2& {\vec{1}}_2\times {\vec{1}}_3\\ {\vec{1}}_3\times {\vec{1}}_1& {\vec{1}}_3\times {\vec{1}}_2& {\vec{1}}_3\times {\vec{1}}_3\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\end{array}\right]$$ $$=[r_1{r}_2{r}_3]\left[\begin{array}{ccc}0& {\vec{1}}_3& -{\vec{1}}_2\\ -{\vec{1}}_3& 0& {\vec{1}}_1\\ {\vec{1}}_2& -{\vec{1}}_1& 0\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\end{array}\right]$$ $$=[{\vec{1}}_1,{\vec{1}}_2,{\vec{1}}_3]\left[\begin{array}{ccc}0& -r_3& r_2\\ r_3& 0& -r_1\\ -r_2& r_1& 0\end{array}\right]\left[\begin{array}{c}{s}_1\\ s_2\\ s_3\end{array}\right]$$ $$={\overrightarrow{{\mathcal{F}}_i}}^T{\mathbf{r}}_1^{\times }{\mathbf{s}}_1$$

Where we express r as a skew-symmetric matrix ${\mathbf{r}}_1^{\times }=\left[\begin{array}{ccc}0& -r_3& r_2\\ r_3& 0& -r_1\\ -r_2& r_1& 0\end{array}\right]$

To adapt to Lie Theory, the ${\mathbf{r}}^{\times }$ is actually denoted as ${\mathbf{r}}^{\wedge }$

Relevant for later (also SO(3))