cross product in cylindrical coordinates Hi i know this is a really really simple question but it has me confused.
I want to calculate the cross product of two vectors
$$
\vec a \times \vec r.
$$
The vectors are given by
$$
\vec a= a\hat z,\quad \vec r= x\hat x +y\hat y+z\hat z.
$$
The vector $\vec r$ is the radius vector in cartesian coordinates.  
My problem is:
I want to calculate the cross product in cylindrical coordinates, so I need to write $\vec r$ in this coordinate system.  
The cross product in cartesian coordinates is
$$
\vec a \times \vec r=-a y\hat x+ax\hat y,
$$
however how can we do this in cylindrical coordinates?  Thank you

The radius vector $\vec{r}$ in cylindrical coordinates is $\vec{r}=\rho\hat{\rho}+z\hat{z}$. Calculating the cross-product is then just a matter of vector algebra:
$$\vec{a}\times\vec{r} = a\hat{z}\times(\rho\hat{\rho}+z\hat{z})\\
=a(\rho(\hat{z}\times\hat{\rho})+z(\hat{z}\times\hat{z}))\\
=a\rho(\hat{z}\times\hat{\rho})\\
=a\rho\hat{\phi},$$
where in the last line we've used the orthonormality of the triad $\{\hat{\rho},\hat{\phi},\hat{z}\}$.
