What are Cross Products and How Do They Relate to Tensors?

In summary, the cross product of two vectors \vec A and \vec B, denoted as \vec A \times \vec B, is equivalent to the vector \vec C = (A_x \hat x + A_y \hat y + A_z \hat z) \times (B_x \hat x + B_y \hat y + B_z \hat z). This can be written as a sum of three terms, each representing the cross product of the unit vectors \hat x, \hat y, and \hat z with the corresponding components of \vec A and \vec B. By using the distributive property and the properties of cross products, the resulting expression can be simplified to (A_x B_y -
  • #1
misogynisticfeminist
370
0
I do not understand something about cross products. Say,

[tex] \vec A\times \vec B=\vec C=(C_x, C_y, C_z) [/tex]

and,

[tex] \vec C=(A_x \hat x+A_y \hat y+A_z \hat z)\times (B_x \hat x+B_y \hat y+B_z \hat z)[/tex]

but why is this equivalent to

[tex] (A_x B_y - A_y B_x)\hat x \times \hat y + (A_x B_z - A_z B_x) \hat x \times \hat z + (A_y B_z - A_z B_y) \hat y \times \hat z [/tex]

?

Can someone show me how do i get this? Preferbly an algebraic method instead of a geometric one, because I am poor at visualizing stuff.
 
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  • #2
Because

[tex] \vec C=(A_x \hat x+A_y \hat y+A_z \hat z)\times (B_x \hat x+B_y \hat y+B_z \hat z)[/tex]

can be written as (using the distributive property)

[tex] \vec C= A_x \hat x \times (B_x \hat x+B_y \hat y+B_z \hat z)[/tex]

+ [tex] A_y \hat y \times (B_x \hat x+B_y \hat y+B_z \hat z)[/tex]

+ [tex] A_z \hat z \times (B_x \hat x+B_y \hat y+B_z \hat z)[/tex]

then [tex] \hat x \times \hat x = \hat y \times \hat y = \hat z \times \hat z[/tex] = 0

and [tex] \hat x \times \hat y[/tex] = - [tex] \hat y \times \hat x[/tex] and so on.
 
  • #3
Hey, thanks, that was helpful, I've gotten it already...

: )
 
  • #4
If you're dealing with vector identites,i've got an advice:learn cartesian tensors.

Daniel.
 
  • #5
another computational way is to use a definition of cross product such as: the cross product of A and B is the determinant of the 3 by 3 matrix with x,y,z (your notation) in the first rwo and the entries of A in the second row, and the entries of B in the third row.
 
  • #6
mathwonk: the matrix idea helped too...

dexter: I apparently haven't even started on vector calc yet, so I don't think i can go into tensors yet.

: )
 
  • #7
do not be afraid of tensors. the dot product is the first tensor we meet. tensors are multilinear as opposed to merely linear. the dot product is bilinear, hence the simplest tensor.


a derivative is a linear map that approximates a function. the second derivative is a bilinear map that approximates the difference between function and its derivative, and so on...


do not be misled by the confusing gobbledygook found here about upper and lower indices as being tensors. that is pablum for people who refuse to learn what tensors are.

there is some truth in it, but it is like saying that a linear map is a matrix. i.e. the matrix is a notational representation of a linear map, not the linear map itself. similarly an array of upper and lower indices is a representation of a tensor and not the tensor itself.

do not be seduced by the "dark side"!
 

1. What is a cross product?

The cross product is a mathematical operation that results in a new vector that is perpendicular to both of the original vectors being multiplied. It is used to calculate the area of a parallelogram or the torque generated by a force.

2. How do you calculate a cross product?

To calculate a cross product, you first need to find the cross product of two vectors by multiplying their magnitudes and the sine of the angle between them. Then, you determine the direction of the resulting vector by using the right-hand rule. Finally, you can find the components of the resulting vector by using the determinant method or the geometric method.

3. What is the difference between a dot product and a cross product?

The dot product and the cross product are both mathematical operations involving vectors. The main difference is that the dot product results in a scalar value, while the cross product results in a vector. The dot product is used to calculate the angle between two vectors or the projection of one vector onto another, while the cross product is used to calculate the area or torque.

4. What are some real-life applications of cross products?

Cross products have many applications in fields such as physics, engineering, and computer graphics. They are used to calculate the torque generated by a force, the magnetic force on a charged particle, and the angular momentum of a rotating object. In computer graphics, cross products are used to determine the direction of the surface normal and to create 3D effects.

5. Are there any limitations to using cross products?

One limitation of cross products is that they can only be calculated for three-dimensional vectors. They also require the vectors to be perpendicular to each other, which may not always be the case in real-life scenarios. Additionally, the magnitude of the resulting vector may be difficult to interpret in some cases, making it less useful for certain applications.

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