A coordinate system can be defined by three perpendicular unit vectors. If the coordinate system is Cartesian, which direction does the $+x$ axis point? To resolve this problem, I define an orientation–a coordinate system that is oriented in a certain direction in three-dimensional space. How many unique orientations are there? Here, a unique orientation is an orientation that cannot be rotated by less than 90 degrees into another orientation —admittedly, 90 degrees is an arbitrary number, but at least it is a start. Two orientations related by less than a quarter of a rotation are considered equivalent in contrast to unique. The idea of a unique orientation can be used to avoid counting a numerous amount of orientations formed by rotations. The answer to the posed question is related to the number of ways that three unit vectors can be arranged, if all three unit vectors are perpendicular. Finally, do the unique orientations form left and right handed coordinate systems? Determining Unique Orientations One way to answer this question is to draw possibilities. If the $+z$ axis points “up” from the perspective of the observer, then the $x$- and $y$- axes are in a plane. For a given placement of the $+x$-axis in this plane, the $+y$-axis must be 90 degrees away from the $+x$-axis. Otherwise, the $x$ and $y$ axes would not be perpendicular. It is important to realize that, for a given position of the $+z$-axis, there are many different directions that the $+x$-axis can point. Since I am considering unique coordinate systems involving a quarter of a rotation, the $+x$ axis can point in one of four directions. Then the $+y$ axis can point in one of two directions. So, if the $+z$-axis points up, then there are 1*4*2=8 ways to uniquely arrange the $+x$ and $+y$ axes. Regardless […]
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