Question

Which of the arrangements of axes in Fig. 3-23 can be labeled “right-handed coordinate system”? As usual, each axis label indicates the positive side of the axis.

Short answer

All the figures except the figure ( e ) are right-handed coordinate systems.

Step-by-step solution

Given information

It has been given that the figures of the coordinate system and each axis label indicate the positive side of the axis.

Vector product

The problem deals with the cross product which is also called a vector product. It is the binary operation on two vectors in three-dimensional oriented Euclidean space. Here the cross product can be used to find out which system is not the right-handed coordinate system.

Formula:

As per the cross-product rule,

$\hat{\mathrm{k}}=\hat{\mathrm{i}}\mathrm{x}\hat{\mathrm{j}}$

To find the arrangements of the axes which can be labeled right-handed coordinate system

For the system to be called a right-handed system, the cross product of the$\hat{\mathrm{i}}\mathrm{and}\hat{\mathrm{j}}$ should be along the $\hat{\mathrm{k}}$. It can be used to verify if all the systems are right-handed. It is found that all the figures except figure (e), give us the same cross product, i.e., along the z-axis or along $\hat{\mathrm{k}}$. So only figure (e) represents the left-handed coordinate system.

Thus, all the figures except the figure are right-handed coordinate systems.