[eli173]

For example, if you calculate the magnitude of a thousand-dimensional vector, you end up with a single scalar. If you calculate the magnitude of a thousand one-dimensional vectors, you'd end up with a thousand scalars. Additionally, if given a thousand-dimensional vector, the ordering is important, whereas if given a thousand one-dimensional vectors, ordering isn't neccesarily something we know about.

The difference is subtle and occasionally pedantic, but can be very important depending on what exactly one is doing.