My Calculus text says that the zero vector is considered parallel to all vectors in the same space, because it's a scalar multiple of any other vector, and scalar multiples are parallel. I can see that certain other results benefit from this convention, such as saying that two vectors are parallel if and only if their cross-product is $0$. On the other hand, we sacrifice other results, such as "vectors parallel to the same vector are parallel to each other."
My question: is the convention taking the zero vector as parallel to all other vectors a universal one? It seems arbitrary to me, and that if we take it as parallel to none, we preserve a more intuitive idea of "parallel", and we can always salvage facts relying on the other convention by throwing in the word "non-zero" as necessary.
If everyone agrees that the zero vector is parallel to all the others, what is it that makes this so useful?
Thanks in advance.
Best Answer
In terms of parallel vectors in calculus, most people would assume that zero vector is parallel to everything. You are really looking at the span of a vector and identifying all those vectors, and zero is always in the span: Two vectors are parallel if their spans are equal. In three dimensions this also has a short and beautiful definition: $u$ and $v$ are parallel if and only if $u \times v = 0$. If you pick a convention, you usually pick the one that gives the nicest definition.
One place the word parallel comes up in for geometry is in "parallel transport" where you move vectors to their parallel counterparts at a different point, and you really want to be able to move the zero vector too.
Of course it depends on the context, in the traditional greek Euclidean geometry context, you only talk about parallel lines, and a zero vector does not define a line. But the ancient greeks also did not talk about vectors at all.