Cross product of polar coordinates

In summary, when using cartesian coordinates, the cross product of basis vectors can be calculated using the expressions i \times j = k, j \times k = i, k \times i = j, j \times i = -k, k \times j = -i, and i \times k = -j. In polar coordinates, the cross product for the vectors r, \theta and z can be written as \vec{i}_{r}\times\vec{i}_{\theta}=\vec{k}, following the right-hand rule. The basis vectors in cylindrical coordinates are also orthogonal and can form a rectangular trihedron, similar to \vec{i},\vec{j}, and \vec{k}.
  • #1
tiagobt
31
0
When using cartesian coordinates, I use the following expressions to calculate the cross product of the basis vectors:

[tex]i \times j = k[/tex]
[tex]j \times k = i[/tex]
[tex]k \times i = j[/tex]
[tex]j \times i = -k[/tex]
[tex]k \times j = -i[/tex]
[tex]i \times k = -j[/tex]

Can I do the same in polar coordinates? How could I write the cross product for the vectors [tex]r[/tex], [tex]\theta[/tex] and [tex]z[/tex]?
 
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  • #2
Yep, the right-hand version is [tex]\vec{i}_{r}\times\vec{i}_{\theta}=\vec{k}[/tex]
and you can complete the cycle from there..
 
  • #3
The cylidrical coordinates are orthogonal,which means that the basis vectors are orthogonal to each other,too.They can be made to form a rectangular trihedron,just like [itex] \vec{i},\vec{j} \ \mbox{and} \ \vec{k} [/itex].

Daniel.
 

1. What is the cross product of polar coordinates?

The cross product of polar coordinates is a mathematical operation that involves two vectors expressed in terms of polar coordinates, which are a type of polar coordinate system commonly used in mathematics and physics. The result of the cross product is a third vector that is perpendicular to both of the original vectors.

2. How is the cross product calculated for polar coordinates?

To calculate the cross product of two vectors in polar coordinates, you first need to convert the polar coordinates to Cartesian coordinates. Then, you can use the standard formula for calculating the cross product of two vectors in Cartesian coordinates: (ax, ay, az) x (bx, by, bz) = (aybz - azby, azbx - axbz, axby - aybx).

3. What are the properties of the cross product in polar coordinates?

The cross product in polar coordinates has several important properties, including being anticommutative (a x b = -b x a) and distributive over addition (a x (b + c) = a x b + a x c). It is also perpendicular to both of the original vectors and has a magnitude equal to the product of the magnitudes of the original vectors multiplied by the sine of the angle between them.

4. What is the significance of the cross product in physics?

The cross product is an important mathematical tool in physics, particularly in mechanics and electromagnetism. It is used to calculate the torque on a rotating object, the magnetic force on a charged particle, and the angular momentum of a system. It is also used in vector calculus and other areas of mathematics.

5. Are there any real-world applications of the cross product in polar coordinates?

Yes, there are many real-world applications of the cross product in polar coordinates. Some examples include calculating the force exerted on a rotating object, determining the direction and magnitude of a magnetic field, and calculating the angular momentum of a system. It is also used in computer graphics and 3D modeling to calculate the orientation of objects in space.

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