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e1 e2 e3
a1 a2 a3
b1 b2 b3
This gives the area of the parallelogram formed by a and b as a vector. It's magnitude is ab sin t where t is the angle between a and b.
So the cross product in 4D, with bases e1, e2, e3, e4, becomes a ternary operation giving the volume of a parallelepiped as a vector.
In 3D, 1D length and 2D area can both be described as vectors. 3D volumes cannot be described as vectors because we cannot associate a unique direction with a volume.
In 4D, 1D length and 3D volume can be expressed as vectors while 4D bulk cannot. 2D areas cannot be associated with a unique direction, but any plane can be specified by giving two non collinear vectors on it. So how do we express 2D area in 4D space? Definitely not as a vector, but is it possible to use a tensor of rank 2?