Cross products for unit vectors in other coordinate systems

In summary: They are looking for a table or formula for cross products such as "phi X rho" and are concerned about the correctness of their computations. They also mention a boundary condition problem in E&M where they had to convert everything into cartesian coordinates, but are wondering if there is a quicker way to do it without going to cartesian coordinates.
  • #1
AdkinsJr
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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.
 
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  • #2
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
 
  • #3
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 [tex]\vec B_{in}=B_o \hat k[/tex] where we were given a surface current density.

[tex]\vec K=K_o\hat\theta[/tex]

and then asked to find the field right oustide applying the boundary conditions for B-fields,

[tex]\vec B_{out}=\mu_o(K_o\hat\theta\times\hat r) + B_o \hat k[/tex]

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 [tex]\hat \theta \times \hat r[/tex] that churns out a Cartesian vector and I ended up with the field just outside as, [tex]\vec B_{out}=\mu_o K_o(sin(\phi)\hat i -cos(\phi)\hat j)+B_o \hat k[/tex]

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 [tex]\hat \theta \times \hat r[/tex] without going to Cartesian?
 
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  • #4
[tex]\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}[/tex]

[tex]\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}[/tex]
 
  • #5
AdkinsJr said:
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.
If the coordinate system is orthogonal, then you do it the way you said (i.e., the same as with cartesian coordinates). But, since the unit vectors generally change direction with position, you need to evaluate the cross product using the unit vectors at the point of interest.

Chet
 

FAQ: Cross products for unit vectors in other coordinate systems

1. What are cross products for unit vectors?

Cross products for unit vectors are mathematical operations used to find the vector perpendicular to two given vectors. This operation is used in various fields of science, including physics and engineering.

2. How do cross products for unit vectors work?

To calculate the cross product of two vectors, you must first take the determinant of a 3x3 matrix. Then, you multiply the vectors by their corresponding cofactors and subtract them. The final result will be the cross product vector in the form of a x b.

3. What is the purpose of using unit vectors in cross products?

Unit vectors have a magnitude of 1 and are used to represent direction. When performing a cross product, using unit vectors ensures that the resulting vector will have a magnitude that is proportional to the sine of the angle between the two vectors.

4. Can cross products be used in other coordinate systems besides Cartesian?

Yes, cross products can be used in other coordinate systems such as cylindrical and spherical coordinates. However, the formula for calculating the cross product may differ depending on the coordinate system being used.

5. What are some applications of cross products for unit vectors?

Cross products for unit vectors have many applications in science and engineering. They are used to calculate torque and angular momentum in physics, and in computer graphics for calculating surface normals and determining the direction of rotation in 3D animations.

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