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.

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 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 convex cube and a 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.

A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point.^{[11]} If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as . In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as: