Azimuthal unit vector ($\hat{\varphi }=(-\sin \varphi ,\cos \varphi ,0)$)

The azimuthal unit vector is tangent to increasing azimuthal angle at fixed cylindrical radius and height. For $r>0$ it equals $(-y/r,x/r,0)$. A counterclockwise circle has tangent displacement $d\mathbf{r}=r\hat{\varphi }d\varphi$. Unlike a fixed Cartesian basis vector, its direction depends on position.