Cross product of two 4-Vectors

Philosophaie
How do you take the cross product of two 4-Vectors?

$$\vec{r} = \left( \begin{array}{ccc}c*t & x & y & z \end{array} \right)$$
$$\vec{v} = \left( \begin{array}{ccc}c & vx & vy & vz \end{array} \right)$$
$$\vec{v} \times \vec{r} = ?$$


Actually it's also well defined non-trivially for 7 dimensions.

Philosophaie
I can do a Triple Product of xyz. I just do not know what to do with the t.

The triple product is:$$v \times r = \bar{x}*(vy*z-vz*y) + \bar{y}*(vz*x-vx*z) + \bar{z}*(vx*y-vy*x)$$

Gold Member
Still not clear what you are trying to accomplish with the 4-vectors here.

Staff Emeritus
Gold Member
2021 Award
Philosophaie, what your asking is how to apply an operation which is not defined for the objects you have produced. It's analogous to me asking you "how do I add two circles together?" It doesn't make any sense to ask the question.

If you give the context under which you want to ask this kind of question, we can probably identify what you actually want to do with your two vectors.

As UltrafastPED said, the vector cross product really only works in three dimensions. In four dimensions, you can form what is called the "wedge product" or "exterior product" of two four vectors, but this object will not be another four-vector. It will be a different geometric object referred to a bi-vector. This object can be defined with a rank-2 anti-symmetric tensor, and it has six components instead of just four.

Philosophaie
I only got 6 dimensions. Check my math:
$$\vec{v} \times \vec{r} = 2*(v_z*c*t-c*z)*\bar{x}\bar{y}+2*(v_y*c*t-c*y)*\bar{x}\bar{z}+2*(v_x*c*t-c*x)*\bar{y}\bar{z}+2*\bar{t}*(\bar{x}*(v_x*y-y-v_y*x)+\bar{y}*(v_z*x-v_x*z)+\bar{z}*(v_x*y-v_y*x))$$

Staff Emeritus