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In mathematics, a 3-dimensional Cartesian coordinate system can be left- or right-handed. These are two possible configurations of the 3 perpendicular axes, which are mirror images of each other.

This distinction arises from the fact that once two perpendicular directions have been chosen for the x and y axes, there are two possible choices for the positive side of z.

The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Alternatively, imagine you are gripping the z axis in a fist in such a way that your fingers curl round from x to y; the direction your thumb points in gives the positive z-axis of the right-handed system for the right hand; and likewise for left.

Another way of determining handedness of axes is as follows:

imagine an ordinary screw lying along the z axis. Now turn it as if by a screwdriver, in the direction from x to y. The direction in which it moves, up or down, is the direction of the positive z axis in a right handed system.

This applies for a screw that is tightened by turning it clockwise, with a so-called right-hand thread. This is the most common kind.

Left-handed on the left, right-handed on the right.

See also chirality, cross product, curl, pseudovector.