Addition of two vectors in diff dimensions

Question

Let's suppose I have two vectors $(1, 2)$ and $(1, 2, 3)$ in $2D$ and $3D$, respectively. And I know vector addition with unequal dimensions is not defined. But if I could make the first vector as $(1, 2, 0)$, is it ok then to add these two vectors?

Answer 1

If it is understood that your ordered pairs are $(x,y)$ while your ordered triples are $(x,y,z)$, then it is perfectly fine to identify $(x,y)$ with the point $(x,y,0)$ and add the two triples together.

Answer 2

You can embed the 2-dim vector in many different ways. So in general the question really makes no sense at all. As several comments indicate, if you state your specific embedding as indicated (and nothing prevents you from doing so) you can do what you've stated. But I would argue that the operation is completely undefined mathematically. It is, at the very least, ambiguous. Baring the embedding trick, I would think that the binary operation "+" is only strictly defined as a map from VXV-->V, and not VXW-->W (or V).