TITLE: A vector is defined to have a magnitude and *a* direction, but the zero vector has no *single* direction. So, how is the zero vector a vector?
QUESTION [11 upvotes]: The popular definition of a vector is 

A vector is an object that has both a magnitude and a direction.

We know that zero vector has no specific single direction.
Then how can it be a vector?

REPLY [8 votes]: What you quote is a reminder, not a definition.
If you are asked to calculate a vector, you know the answer shouldn't be a single number.
The zero vector disagrees with that reminder, and that is a useful caveat to learn.
When vectors are first introduced, they might be treated as arrows and explained with pictures in a way that emphasizes that direction matters.
But if you actually get to give a definition of a vector in linear algebra (an element of a vector space), it looks different.
The direction is of a vector is not a canonically defined object, just a useful way of thinking.
Consider two possible definitions in $\mathbb R^n$:

The direction of a vector $x\in\mathbb R^n$ is a vector $v\in S^{n-1}$ so that $x=tv$ for some $t\in\mathbb R$, $t\geq0$.
The direction of $x$ is the unit vector $x/|x|$.

They only disagree on the zero vector, and both definitions are reasonable.
Whether the zero vector has any or no direction is an irrelevant choice.
If you are on a space without a norm, you can define a direction as an equivalence class.
Sometimes it can be useful to think that $v$ and $-v$ have the same direction, sometimes not.
Going from a vector space to the corresponding space of directions is essentially projectivization.
How you should define a direction depends on what you want to do with it.
At a general level with no application in sight I would say that there simply is no canonical definition.
Sometimes it's useful to say $1/0=\infty$, sometimes it's better to leave $1/0$ undefined.
If these concepts are unfamiliar or weird to you, the message is even clearer:
The concept of direction is a tricky one to define.
I think it's often best left as a heuristic term instead of a technical one.