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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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    One can actually set things up so that you can (iteratively) compute $\sqrt{x^2+y^2}$ (probably) more cheaply than squaring $x$ and $y$ and rooting their sum. If you're interested in that, you can ask a separate question and I'll be happy to elaborate.2011-05-04

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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.

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    @Emre: If he already thinks square roots are expensive, then certainly trigonometric functions would $b$e even more costly to evaluate.2011-05-04