Question

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.

Short answer

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

Step-by-step solution

Step 1. Given Information

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.

Step 2. Determine right-handed systems and left-handed systems 

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.

Step 3. Find all of the \(x, y, z\) labelings of the axes in the diagram

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