When defining a cartesian coordinate system in 3 dimension, once two of its axes are specified, we can define a plane where the origin and both axes lie. The third axis will point away from that plane on either side. Depending on which side of the plane we choose to place the third axis in, we have a left-handed system or a right-handed system.
In the case where the axes are required to be orthogonal, defining two axes restricts the third axis to lie on a line. The only thing to be determined, now, is the direction.
Just as the left hand and the right hand are mirror images of each other, left handed systems and right handed systems represent mirror images of each other.
It is possible to transform any right handed system into any other right handed system with only rotation, shearing and non-negative scaling transformations. The same happens with left handed systems. However it is not possible to transform a left-handed system into a right-handed system, or the other way around using only those transformations.
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