# Is this 4-Vector cross product calculation correct?

• I
• Philosophaie
In summary, The concept of a 4-vector cross product does not exist. The closest equivalent is the wedge product between two 1-forms in exterior algebra. To understand how to perform the wedge product, the concept of exterior algebra should be researched. Additionally, the purpose of needing a generalization of the cross product should be identified in order to point towards an appropriate direction. The Maxwell equations in 4 dimensions can be solved using the tensor formalism of special relativity, with the field tensor being represented by the antisymmetric rank 2 tensor F. Finally, the cross product can be used to solve for some Maxwell equations, with the gradient operator represented by the partial derivatives of space and time.
Philosophaie
Could someone tell me if this 4-Vector cross product is correct:

i j k t
dx dy dz 1/c*dt
Ex Ey Ez Et
=[(dy(Ez)-dz(Ey))-(dy(Et)-1/c*dt(Ey))+(dz(Et)-1/c*dt(Ez))]*i
-[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*j
+[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*k
-[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*t

There is no such thing as a 4-vector cross product. The closest thing you will get is the wedge product between two 1-forms (or rather, its dual), which is a 2-form.

Auto-Didact
How do you do the wedge product between 2 1-forms.
I searched for Wedge Product. I got Exterior Algebra. Here are some Equations could someone explain the "^" operator to me.
?
where k=4

?

I believe a better path is to identify your purpose, what do you need a generalisation of the cross product for? Once we have established that, we may be able to point you in an appropriate direction.

I am trying to solve Maxwell Equations in 4 dimensions.

Maxwell’s equations are already in 4 space-time dimensions. Their appearance in the tensor formalism of special relativity takes the form ##\partial_\mu F^{\mu\nu} = J^\nu## and ##\epsilon^{\mu\nu\sigma\rho}\partial_\nu F_{\sigma\rho}=0##, where F is the antisymmetric rank 2 field tensor.

To solve for for some Maxwell Equation it takes the cross product:
dx = partial d/dx
dy = partial d/dy
dz = partial d/dz
dt = partial d/dt

grad X H(r,t) = dt(D(r,t)) + J

Last edited:

## What is a 4-Vector Cross Product?

A 4-vector cross product is a mathematical operation used in the study of special relativity and in theoretical physics. It involves taking the cross product of two 4-vectors, which are mathematical objects with four components that represent physical quantities such as position, velocity, and momentum.

## How is a 4-Vector Cross Product calculated?

To calculate the 4-vector cross product, you first need to take the cross product of the first three components of each 4-vector, similar to how a regular cross product is calculated. Then, for the fourth component, you take the dot product of the first three components and subtract it from the dot product of the last three components. This results in a new 4-vector with four components.

## What is the significance of the 4-Vector Cross Product in physics?

The 4-vector cross product is used in special relativity to calculate physical quantities such as force, momentum, and energy. It is a crucial mathematical tool in understanding the behavior of objects moving at high speeds and in different frames of reference.

## Are there any real-world applications of the 4-Vector Cross Product?

Yes, the 4-vector cross product has various applications in physics, particularly in the fields of particle physics and cosmology. It is also used in practical applications such as GPS technology and spacecraft navigation.

## Can the 4-Vector Cross Product be extended to higher dimensions?

Yes, the concept of the 4-vector cross product can be extended to higher dimensions, such as 5 or 6 dimensions. However, the calculations become increasingly complex and are not commonly used in physics.

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