>>16752569 (OP)
>>16752623
>>16752626
in calc 1 you learn
vector x vector = vector (cross product
vector * vector = scalar (dot product)
the cross product retains the vector's dimensionality; the dot product collapses the vector's dimensionality. it would seem possible there exists another product that expands the vector's dimensionality.
vector#vector = matrix (dyadic product)
it's not just historical.