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Does the method for computing the cross-product change for left handed coordinates?
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For a cross product, using a counter-example the left-hand rule is proved wrong. The unique rule for a cross product is the right-hand rule. Coordinate systems have handedness, while a cross product is frame-indifference.
In this case, right-handedness is defined as any positive axis (x, y, or z) pointing toward the viewer. Left-handedness is defined as any positive axis (x, y, or z) pointing away from the viewer.
To convert a right-handed vector (assuming that z is up) to left-handed vector (Unity coordinate system), simply swap y and z coordinates.
For left-handed coordinates the left thumb points along the z axis in the positive direction and the curled fingers of the left hand represent a motion from the first or x axis to the second or y axis. When viewed from the top or z axis the system is clockwise.
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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