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?

Solution 1:

You are right to question this definition. It suggests that every vector is associated with a unique direction. This is almost true, with the sole exception of the zero vector, which cannot sensibly be said to have any direction. Unfortunately, a more accurate definition

“a vector is an object that has both a magnitude and a direction, with the sole exception of the zero vector, which cannot sensibly be said to have any direction”

is considerably less snappy. When you get to more advanced treatment of vectors, the lack of a direction for the zero vector is made quite explicit.

Solution 2:

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 $\mathbb R^n$:

1. The direction of a vector $x\in\mathbb R^n$ is a vector $v\in S^{n-1}$ so that $x=tv$ for some $t\in\mathbb R$, $t\geq0$.
2. 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.

Solution 3:

Yes, it should have a direction, but I see no place in the quoted definition where it's stated that it should have only a direction.

The idea is simple. This indefiniteness in direction happens only for the trivial (zero) vector. It is a vector that does nothing. And a point can be oriented indefinitely, unlike a line segment. Furthermore, we have to include it to make our computation with vectors become very like normal arithmetic (where although we can't divide by $0,$ we still have to include it). If we do not include it, we inconvenient ourselves, so we include it.

Later, you'll learn that mathematicians define vector differently. In that case, the zero vector automatically becomes a vector without questions of definiteness.

Solution 4:

$0 \cdot (a,b,c)$ (for arbitrary $a$, $b$, and $c$) has a magnitude ($0$) and a direction $(a,b,c)$. It is a vector.

$0 \cdot (d,e,f)$ (for arbitrary $d$, $e$ and $f$) has a magnitude ($0$) and a direction $(d,e,f$). It is a vector.

These happen to be the same vector.

So what?