Editing Cartesian coordinate system (section)

===In three dimensions===
[[File:Cartesian coordinate system handedness.svg|left|thumb|Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.]]
[[File:Right hand cartesian.svg|thumb|Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes]]

Once the ''x''- and ''y''-axes are specified, they determine the [[line (geometry)|line]] along which the ''z''-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems, which result are called 'right-handed' and 'left-handed'.<ref>{{harvnb|Anton|Bivens|Davis|2021|p=[https://books.google.com/books?id=001EEAAAQBAJ&pg=PA657 657]}}</ref> The standard orientation, where the ''xy''-plane is horizontal and the ''z''-axis points up (and the ''x''- and the ''y''-axis form a positively oriented two-dimensional coordinate system in the ''xy''-plane if observed from ''above'' the ''xy''-plane) is called '''right-handed''' or '''positive'''.

[[File:3D Cartesian Coodinate Handedness.jpg|thumb|3D Cartesian coordinate handedness]]

The name derives from the [[right-hand rule]]. If the [[index finger]] of the right hand is pointed forward, the [[middle finger]] bent inward at a right angle to it, and the [[thumb]] placed at a right angle to both, the three fingers indicate the relative orientation of the ''x''-, ''y''-, and ''z''-axes in a ''right-handed'' system. The thumb indicates the ''x''-axis, the index finger the ''y''-axis and the middle finger the ''z''-axis. Conversely, if the same is done with the left hand, a left-handed system results.

Figure 7 depicts a left and a right-handed coordinate system. Because a three-dimensional object is represented on the two-dimensional screen, distortion and ambiguity result. The axis pointing downward (and to the right) is also meant to point ''towards'' the observer, whereas the "middle"-axis is meant to point ''away'' from the observer. The red circle is ''parallel'' to the horizontal ''xy''-plane and indicates rotation from the ''x''-axis to the ''y''-axis (in both cases). Hence the red arrow passes ''in front of'' the ''z''-axis.

Figure 8 is another attempt at depicting a right-handed coordinate system. Again, there is an ambiguity caused by projecting the three-dimensional coordinate system into the plane. Many observers see Figure 8 as "flipping in and out" between a [[wikt:convex|convex]] cube and a [[wikt:concave|concave]] "corner". This corresponds to the two possible orientations of the space. Seeing the figure as convex gives a left-handed coordinate system. Thus the "correct" way to view Figure 8 is to imagine the ''x''-axis as pointing ''towards'' the observer and thus seeing a concave corner.
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