In the derivation of numerous equations of motion a double cross product or vector triple product occurs. This can be resolved using two cross product operations or by using the following identity:

Notice that the two terms on the right hand side of the equation do not have a symbol before the vector. That is because the operations within the parentheses result in scalars that multiply each term of the vector. A dot product where the gradient (∇) operates on another vector is known as the gradient.

The curl of the gradient of a scalar, F, is always zero:

The divergence of a curl is always zero:

Examples of typical cross product problems, as well as applications illustrating the usefulness of the cross products can be found in the remainder of the section.