I admit that
Cross[{0,1,0,0},{0,0,1,0},{0,0,0,1}]= - {1,0,0,0} (1)
applied to four 4-dimensional basis vectors
e1={0,1,0,0}, e2={0,0,1,0}, e3={0,0,0,1}], and e4={1,0,0,0}
can successfully lead us to a calculation of the volume V
V = Cross[v1,v2,v3].v4 (*nner product of Cross[v1,v2,v3] with v4*)
of a parallelohedron made up of four 4-dimensional vectors
vi = {xi,yi,zi,wi} = e1 xi + e2 yi +e3 zi +e4 wi (i=1,2,3,4)
represented on the four 4-dimensional orthonormal unit basis vectors
e1={0,1,0,0}, e2={0,0,1,0}, e3={0,0,0,1}, and e4={1,0,0,0}
On the other hand, however, the most natural extension of a well known cross product
Cross[{1,0,0},{0,1,0}]={0,0,1}
would be
Cross[{0,1,0,0},{0,0,1,0},{0,0,0,1}]= {1,0,0,0}, not -{1,0,0,0}. (2)
I will be happy if someone kindly explains me why Cross of 3 unit vector be given by (2).
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