# Cross products for unit vectors in other coordinate systems

Tags: coordinate, cross, products, systems, unit, vectors
 P: 124 I am a bit confused often when I have to compute cross products in other coordinate systems (non-Cartesian), I can't seem to find any tables for cross products such as "phi X rho." in spherical I think that these unit vectors are considered to be "perpendicular," so would phi X rho just be "+/- theta," in general? Typically when I'm doing problems in E&M it takes me a while to convince myself that my computations are correct in terms of direction and it's just frustrating. On an exam I need to just know what the cross products are quick. My hang up is just that they vary from place-to-place.
 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 6,318 It's not clear why you are trying to calculate cross products in non-cartesian coordinate systems. AFAIK, the vector cross product is defined only for cartesian coordinates, and then only for 3-dimensional (and 7-dimensional) coordinates. http://en.wikipedia.org/wiki/Cross_product
 P: 124 3-dimensional which would include spherical and cylindrical correct? An example would be a boundary condition problem we had in one of our homeworks (for E&M). We were given the B field just inside a spherical shell as $$\vec B_{in}=B_o \hat k$$ where we were given a surface current density. $$\vec K=K_o\hat\theta$$ and then asked to find the field right oustide applying the boundary conditions for B-fields, $$\vec B_{out}=\mu_o(K_o\hat\theta\times\hat r) + B_o \hat k$$ You can see that the field just inside is in Cartesian and K is in spherical. I converted everything into Cartesian coordinates by writing out the determinant matrix for $$\hat \theta \times \hat r$$ that churns out a Cartesian vector and I ended up with the field just outside as, $$\vec B_{out}=\mu_o K_o(sin(\phi)\hat i -cos(\phi)\hat j)+B_o \hat k$$ He did not object to my expression, it's pretty straight forward, but it was more tedious computationally because of the determinant matrix and everything.... I remember he had a different expression though in spherical coordinates and I can't really ask him at the moment I just have a test coming and was thinking it would be nice to deal with vectors more quickly than I do because I don't have a lot of insight when it comes to using these basic vector operations in other coordinate systems. I guess my question is, can I compute $$\hat \theta \times \hat r$$ without going to Cartesian?
 P: 686 Cross products for unit vectors in other coordinate systems $$\begin{matrix} \times & \hat{q}_1 & \hat{q}_2 & \hat{q}_3 \\ \hat{q}_1 & \vec{0} & +\hat{q}_3 & -\hat{q}_2 \\ \hat{q}_2 & -\hat{q}_3 & \vec{0} & +\hat{q}_1 \\ \hat{q}_3 & +\hat{q}_2 & -\hat{q}_1 & \vec{0} \\ \end{matrix}$$ $$\begin{matrix} system & \hat{q}_1 & \hat{q}_2 & \hat{q}_3 \\ cartesian & \hat{x} & \hat{y} & \hat{z} \\ cylindrical & \hat{r} & \hat{\theta} & \hat{z} \\ spherical & \hat{\rho} & \hat{\phi} & \hat{\theta} \\ \end{matrix}$$