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

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?

Umm... the zero vector has magnitude $0$ AND IS IN THE DIRECTION (1,0,0)... or in the direction $(a,b,c)$... or in the direction ... All of these are vectors that happen to have zero magnitude. Each of these have "a" direction. Nobody said it had to be unique. — David G. Stork, Aug 31 '19 at 06:23
Relevant: https://math.stackexchange.com/questions/1896784/do-all-vectors-must-have-a-direction-and-magnitude — Theo C., Aug 31 '19 at 06:32
Nice. I didn't notice this. In this case, "a direction" then is "at least 1 direction" rather than "exactly 1 direction". Or you can say "exactly 1 direction" and then define the zero vector to be in whatever direction you want. — , Aug 31 '19 at 07:04
Note that there is a completely different way to define a vector that does not use the concept of direction at all. In this alternative view, a vector space is defined first as a set with operations that have certain properties, and then any element of that set is called a vector. It’s possible some of what you’ve read about the zero vector or zero vectors has this or another definition of vector in mind. — Todd Wilcox, Aug 31 '19 at 17:47
A snake has one head, but in the rare occasions when it has two, it is nevertheless a snake. — Mirko, Aug 31 '19 at 22:57
@Micah The direction of a vector is the measure of the angle it makes with a horizontal line . — hanugm, Sep 01 '19 at 10:07

score 12 · Answer 1 · answered Aug 31 '19 at 07:18

12

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.

answered Aug 31 '19 at 07:18

John Bentin

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Actually, the zero vector has infinitely many directions. The problem is with assuming that the definition claims uniqueness of directions, which it doesn't. One should train oneself not to read much more than is stated in mathematics. – Allawonder Aug 31 '19 at 07:29
@Allawonder : Clearly we disagree. But I agree with the last, general, point in your comment. – John Bentin Aug 31 '19 at 07:35
Clearly. But the question is, which is correct (at least we'll agree that one must be correct, no?)? Is it the case that it simply makes no sense to talk about directions for points, or simply that we can't talk about definite directions for them? I think it's the latter -- imagine a working ceiling fan. – Allawonder Aug 31 '19 at 07:55
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@Allawonder Neither has to be correct. Direction is not a canonically defined object, just a useful way of thinking. Consider two possible definitions: (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 $t\in\mathbb R$. (2) The direction of $x$ is the unit vector $x/|x|$. They only disagree on the zero vector, and both definitions are reasonable. – Joonas Ilmavirta Aug 31 '19 at 19:10
@JoonasIlmavirta Are you then saying that there is another possibility from The idea of direction makes no sense for the zero vector and The idea of direction makes sense for the zero vector, but the direction cannot be definite and The idea of direction makes sense for the zero vector and it is definite, the latter of which (hopefully) everyone agrees is out of the game? If you have other such possibilities, it would be interesting to see. Otherwise, the two possibilities on ground are obviously mutually exclusive. – Allawonder Aug 31 '19 at 20:01
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@Allawonder What I'm saying is that there need not be right and wrong here. To me "What is the direction of the zero vector?" is like "What is the best flavor of ice cream?" in that even if we can list all possibilities there is no natural way to decide what is the correct answer. You seemed to assume that there is a single correct answer. I disagree on that. This is not not a matter of logic but of ways of thinking. I have no preferred choice myself (except with ice cream). I wrote something in a separate answer. – Joonas Ilmavirta Aug 31 '19 at 20:20
@JoonasIlmavirta If there need not be right or wrong, then we're talking opinion, not mathematics. But this is mathematics. Furthermore, the question is not What is the direction of the zero vector? The question is whether it makes sense to talk about direction or it doesn't. One of these possibilities must be the case -- either it makes sense or not. I'm afraid you've completely misunderstood me -- I don't assume such a thing as you think I do. Indeed, on the contrary, I made an answer below where I stated my points more elaborately (which see please). Clearly, it's a matter of logic. – Allawonder Aug 31 '19 at 20:41
@Allawonder I have to add that opinion is very important in mathematics, and so is debating what definitions are useful. Whether something makes sense or not is a good question but often not a binary and fully logical one. As a professional mathematician I have started to see things very differently from puristic "math is all about true and false statements" point of view. We appear to disagree, but I will end the discussion on my part here. I might answer if there is a separate question on the topic. – Joonas Ilmavirta Aug 31 '19 at 20:49
@JoonasIlmavirta Opinion is important of course, but my point was that the question of whether It doesn't make sense to talk about directions for zero vectors or It makes sense, just that the directions are indefinite is right is not a matter of opinion, but of logic since they both exclude each other. The question between you and me is thus not of what makes sense, but of the fact that one of the statements has to be true. – Allawonder Aug 31 '19 at 21:10
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"It suggests that every vector is associated with a unique direction." It absolutely does not. In mathematics, "X has a Y" does not imply "X has an unique Y". If we want to say the latter, we say the latter. Otherwise you couldn't say things like "any integer $>1$ has a prime factor", and it would be ridiculous. – fkraiem Sep 01 '19 at 02:49
@fkraiem : In English, it is paradoxical, if it makes sense at all, to say that one thing is pointing in two directions: the word direction implies uniqueness. As you say, there is no such implication in the idea of having a prime factor. – John Bentin Sep 01 '19 at 06:28
@JohnBentin It depends on the thing you're talking about. What's the unique direction in which a spinning fan points, eh? ;) – Allawonder Sep 02 '19 at 04:06
@Allawonder : Forwards (downstream of the airflow) along its axis of spin—so (normally) downwards in the case of a ceiling fan—whether it is spinning or not. – John Bentin Sep 02 '19 at 08:28
@JohnBentin Curious. I'm talking of a spinning fan. The object here is not a fan, but a spinning fan. – Allawonder Sep 02 '19 at 08:36
The simple definition of a vector as an object with size and direction is a very simple one which does not transfer to higher levels of understanding. – Tom Sep 03 '19 at 18:35

Joonas Ilmavirta · Answer 2 · 2019-08-31T20:24:21.570

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.

score 5 · Answer 3 · answered Aug 31 '19 at 07:00

5

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.

answered Aug 31 '19 at 07:00

Allawonder

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Re: "I see no place in the quoted definition where it's stated that it should have only a direction": So then, do you also believe that a nonzero vector can have multiple definitions? – ruakh Aug 31 '19 at 16:12
@ruakh I don't understand your question. But before your reply, perhaps let me reiterate my point. The definition given by OP does not say anything about uniqueness of directions, so it does not exclude a vector which might have indefinitely many directions. – Allawonder Aug 31 '19 at 20:04

score 1 · Answer 4 · answered Aug 31 '19 at 06:40

$0 \cdot (a,b,c)$ (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?

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?

4 Answers4

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?