cross product in cylindrical coordinates

Question

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

Answer 1

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 $\{\hat{\rho},\hat{\phi},\hat{z}\}$.