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).
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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.