  In textbooks and historical literature, the cross product has been defined
only in 2-dimensional and 3-dimensional Euclidean spaces and the cross product
of only two vectors has been defined only in the high dimensional Euclidean
space whose metric matrix is the unit matrix. Nobody has given a universal
definition for any number of vectors in high dimensional spaces whose metric
matrices are the unit matrices. In fact, we can also define the cross product
of m vectors in an n-dimensional space, where n and m can take any positive
integers larger than 1 and m must not be larger than n. In this paper, we give
the definition of the cross product of m vectors in n-dimensional spaces whose
metric matrices are any real symmetric or Hermitian matrices, and put forward
two theorems related to matrices, so as to perfectly explain the geometric
meaning of the cross product of vectors in high dimensional spaces.
Furthermore, the length of the cross product represents the m-dimensional
volume of the parallel polyhedron spanned by the m vectors, and the absolute
value of each component of the cross product represents each component of the
volume in different directions. Specially, in the high dimensional Euclidean
and unitary space where the metric matrix is the unit matrix, this
decomposition of the volume still satisfies the Pythagorean theorem, and the
cross product of n vectors in an n-dimensional space is the determinant of the
square matrix which is formed with these n vectors as row or column vectors. We
also explain the geometric meaning of the determinants of the metric matrices
and their sub-matrices, which is also useful for understanding the invariant
volume elements in high dimensional spaces and their subspaces in the
differential geometry.
