Does the method for computing the cross-product change for left handed coordinates?
The formula for the cross product of the vectors (x1, x2, x3) and (y1, y2, y3) is
z1 = x2 * y3 - x3 * y2
z2 = x3 * y1 - x1 * y3
z3 = x1 * y2 - x2 * y1
It is designed in a way that the three vectors x, y and z in the given order have the same handedness as the coordinate system itself. This property does not depend on the handedness of the coordinate system -- for a left-handed coordinate system the vectors fulfil the left-hand rule. You don't need to change anything about the formula.
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In left-handed coordinates, the cross product is the same magnitude simply pointed the other way. The easiest way then that I know of to convert from a right-handed coordinate system's cross-product, and a left-handed coordinate system's cross-product, is to take the components of the right-handed cross-product and reverse the signs.
Yes, the magnitude of the cross-product for left-handed coordinates is negative that for right-handed coordinates.
来源：https://stackoverflow.com/questions/4820400/does-the-method-for-computing-the-cross-product-change-for-left-handed-coordinat