Parallelepiped used in a sentence

Parallelepiped in a sentence as a noun

Of the parallelepiped generated by the columns of the matrix.

Not a very useful addition but hypercube is to cube as parallelotope is to parallelepiped.

Then the determinant is "only" the volume of the image of the unit hypercube under the linear transformation, which is where the parallelepiped comes in.

This drives in part the parallelepiped interpretation that the article is giving; in fact if you have a full set of eigenvectors then any volume can be made up out of little boxes made of the eigenvectors, and so any hyper-volume must transform under the linear map by multiplying by the determinant.

How can you clarify anything by taking your vectors to be infinitesimal rotations, whose cross product gives a third infinitesimal rotation, which carries information about the non-commutativity of the [non-infinitesimal] rotations?It is one thing that the cross product does, but I don't see how it explains, for example, what the cross product has to do with calculating volumes of parallelepipeds, or torques.