The moment of inertia is typically associated with principal axes, but it can also be computed for any random axis. When an arbitrary axis is under consideration, the moment of inertia is determined by integrating the mass distribution of the object along that specific axis. It is crucial in applications like the design of machinery, where components rotate about various axes, and balance and stability are essential.

In this scenario, the perpendicular distance between the chosen arbitrary axis and the mass distribution being considered is derived from the cross-product of the unit vector, which defines the direction of the axis and the position vector for the mass element.

When a dot product operation is performed, and the brackets are expanded, it results in the expression for the moment of inertia. Utilizing the definitions of the moment of inertia and the product of inertia along various axes, the moment of inertia along an arbitrary axis can be generalized.

Suppose the inertia tensor is defined with respect to the XYZ axes. In that case, calculating the moment of inertia about an arbitrary axis becomes possible if the direction cosines of the axis are known. This approach allows for a more comprehensive understanding of the moment of inertia, beyond the limitations of considering only principal axes.