Do four dimensional vectors have a cross product property?

Question

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$

Where :
$ \vec A = (A_i; A_j; A_k)$
$ \vec B = (B_i; B_j; B_k)$
$ \vec C = (C_i; C_j; C_k)$

$C_i = \left|\begin{matrix}A_j&A_k\\B_j&B_k\end{matrix}\right|$ $C_j = \left|\begin{matrix}A_k&A_i\\B_k&B_i\end{matrix}\right|$ $C_k = \left|\begin{matrix}A_i&A_j\\B_i&B_j\end{matrix}\right|$

But what about if we have four dimensional vectors?
Is it possible to make cross product of four dimensional vectors?
If it is possible, then tell me when it can be possible?

Let say we have two vectors:
$ \vec A = (A_i; A_j; A_k; A_l)$
$ \vec B = (B_i; B_j; B_k; B_l)$

Then how to compute a cross product of this two vectors? Will it again vector? $$ \vec A \times \vec B = \vec C$$

$ \vec C = (C_i; C_j; C_k; C_l)$
Then how to compute those coordinates?

We know that only square matrices have a determinant property!
In this case it might not be correct if we will wright...
$\color{red} {\text { $C_i = \left|\begin{matrix}A_j&A_k&A_l\\B_j&B_k&B_l\end{matrix}\right|$} C_j = \left|\begin{matrix}A_k&A_i&A_l\\B_k&B_i&B_l\end{matrix}\right| C_k = \left|\begin{matrix}A_i&A_j&A_l\\B_i&B_j&B_l\end{matrix}\right| C_l = \left|\begin{matrix}A_i&A_j&A_k\\B_i&B_j&B_k\end{matrix}\right|}$

So tell me how to solve this problem?

Answer 1

While it is feasible to compute a cross-product in four dimensions, the cross-product only has the orthogonality property in three and seven dimensional spaces. You should consider instead looking at Gram-Schmidt Orthogonalization to find orthonormal vectors.

http://www.math.hmc.edu/calculus/tutorials/gramschmidt/

Answer 2

Here is an article that precisely formulates and demonstrates the non-existence of a cross product in Euclidean spaces of dimensions other than $3$ and $7$: http://www.jstor.org/stable/2323537. If you can't access the article through the link, the article reference is

Cross Products of Vectors in Higher Dimensional Euclidean Spaces

W. S. Massey

The American Mathematical Monthly

Vol. 90, No. 10 (Dec., 1983), pp. 697-701.

Since it's in the Monthly, it should be pretty readable to a well-prepared undergraduate.