Double cross product in 2D

Question

Hello I have a question about a double cross product, appearing in centrifugal force \begin{align*} \mathbf{F}_{centrifugal} = -m \boldsymbol{\omega} \times [\boldsymbol{\omega} \times \mathbf{r}] \, . \end{align*} Is there an option to write this for 2D (maybe using bivectors?)?

Answer 1

We can use the back-cab rule as following$$\mathbf{F}_{centrifugal} = -m \boldsymbol{\omega} \times [\boldsymbol{\omega} \times \mathbf{r}]=-m((\boldsymbol{\omega}\cdot\boldsymbol{r})\boldsymbol{\omega}-|\boldsymbol{\omega}|^2\boldsymbol{r})=m|\boldsymbol{\omega}|^2r-m(\boldsymbol{r}\cdot \boldsymbol{\omega})\boldsymbol{\omega}$$