Is there a way to get the result of a cross product to be normalized after just a cross action, i.e. without doing after the cross
v/|v|? (the vectors involved are not normalized, but they are orthogonal).
If two vectors are orthogonal, then the length of their cross product is the product of their lengths. So if by "normalized" you mean length $1$, just divide by the product of the lengths of the two vectors.