A linear map $f:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ is defined as $\overrightarrow{u}\times (\overrightarrow{r}\times \overrightarrow{u})$ where $\overrightarrow{u}$ is a unit vector. What would its geometry look like?

I know that I can rewrite this map as $(\overrightarrow{u}\cdot \overrightarrow{u})\overrightarrow{r}-(\overrightarrow{u}\cdot \overrightarrow{r})\overrightarrow{u}=\overrightarrow{r}-(\overrightarrow{u}\cdot \overrightarrow{r})\overrightarrow{u}$

However I am not sure what to do from here.

I know that I can rewrite this map as $(\overrightarrow{u}\cdot \overrightarrow{u})\overrightarrow{r}-(\overrightarrow{u}\cdot \overrightarrow{r})\overrightarrow{u}=\overrightarrow{r}-(\overrightarrow{u}\cdot \overrightarrow{r})\overrightarrow{u}$

However I am not sure what to do from here.