I have been meaning to find the difference between $2$ $n$-dimensional vectors. I am looking for a convex function (anti-commutative) that finds a signed difference between the two vectors (in the range $-1$ to $1$).
Taking $L_2$ norm certainly provides the distance between the two but with a strictly positive range.
Cross product seems like a possible direction but cross product of two $n$-dimensional vectors results in a tensor of rank $n-2$ whose sign is computed by noting the permutation of the indices of the tensor. However, I could not find any details in this direction.
Can anyone help in determining a function with a simpler computation of a signed difference of $2$ $n$-dimensional vectors?
Thanks in advance!