vector operations between a two-dimensional and a three-dimensional vectors If you have a two-dimensional and a three-dimensional vectors, are vector operations (like dot product, cross product, addition, etc) defined between these two vectors? I though that since two-dimensional vectors can be written as three dimensional vectors with third component being zero, this is possible.

Depends how nitpicky you want to be. If you can embed the two-dimensional vector into a subspace of the three-dimensional space, then of course there is a natural way to extend it. Depends on what you want to do!
