Find all of the \(x, y, z\) labelings of the axes in the diagram that follows. Determine which of your labelings are right-handed systems and which are left-handed systems.

The right-handed system is defined as if the index finger of the right-hand points in the positive x direction and the middle finger of the right-hand points in the positive y direction, then the right thumb will naturally point in the positive z direction.

The Left-handed system is defined as any positive axis \(x, y, z)\) pointing away from the viewer.

When in the diagram the z-axis is vertical and increases as you go up and the y-axis increases from left to right, and the x-axis should be interpreted as pointing straight out from the page increasing as it comes toward you then the diagram is an example of a right-handed system because it obeys the following “right-hand rule”: If the index finger of the right-hand points in the positive x direction and the middle finger of the right-hand points in the positive y direction, then the right thumb will naturally point in the positive z direction.

When in the diagram the z-axis is vertical and increases as you go up and the x-axis increases from left to right, and the y-axis should be interpreted as pointing straight out from the page increasing as it comes toward you then the diagram is an example of a left-handed system because it obeys the following “left-hand rule”: If the index finger of the left-hand points in the negative x direction and the middle finger of the left-hand points in the positive y direction, then the right thumb will naturally point in the positive z direction.

The \(x, y, z\) labelings of the axes in the diagram are as follows and the labelings of right-handed systems and left-handed systems