What are the technical reasons that we must define vectors as "arrows" and carefully distinguish them from a point?

Question

I thought I'd bring this question to math.SE, as it could spark some interesting discussion.

When I first learned vectors - a long time ago and in high school - the textbook and teachers would always introduce them as a "quantity with both a magnitude and direction"

This definition always seemed to irk me. It seems to favor the "polar" definition of a vector and then teach "but it has Cartesian components [x_0, x_1 ... x_n] like so"

I felt that I reached a personal breakthrough when I realized that "A vector is a higher dimensional generalization of a 'number' or 'value' or 'quantity' "

This has been how I've always thought of them.

For example, the number five can be thought of as a one dimensional vector - <5>.

< . . . . . 0---------> . . .>
            0         5

Just like the vector <5,5> is a two dimensional analog of this notion. It just so happens that the circumstances change in subtle ways - (the ability to have a 'distance' or 'absolute value' different from either of the components and the notion of a direction when graphed).

Also, it seems that every teacher is very very careful to distinguish a vector from a point. However, this seems trivial in concept, as we wouldn't distinguish "the position '5' on a number line" from "the value 5" or "distance from zero to 5"

Why is the 'magnitude and direction' definition favored?

Answer 1

If you have a space with no origin then you can still have points that are placed relatively to each other, and differences between these points can be vectors. This is related to the distinction between affine and euclidean.

Answer 2

Why is the 'magnitude and direction' definition favored?

It's not. Mathematicians generally think of vectors as abstract objects, which live in what is called a vector space. Essentially, vectors are things we can add together and which we can multiply by scalars. The treatment of vectors as arrows is considered simpler to teach by many, but breaks down for more complicated examples. For example, real numbers form an infinite-dimensional vector space over the rational numbers, yet there's no clear way to think of them as "arrows".

Answer 3

You're asking several related questions, so I'm just going to address the one in the title. The short story is that we distinguish between vectors and points because you can add vectors, but you can't add points.

I hear you saying "Of course I can add points! Look, I just take the components and add them, just like that." But where did you get those components from? You had to choose some coordinate system first. In particular, you had to choose an origin, then you chose axes coming out of that origin. When you do this, you aren't really adding points anymore; you're adding vectors! Namely, the vectors starting from the origin and ending at your points. If you choose a different origin, you get a different notion of addition this way.

This has real significance. For example, it does not make any sense to add two energy levels of some physical system because there is no physically meaningful "origin" for energy levels; you can take all of the energy values you care about and add the same constant to each of them and it doesn't make any difference physically. It does, however, make sense to subtract two energy levels; a difference between energy levels really is a real number (that is, a one-dimensional vector).

The mathematical terms to look up here are vector space and affine space.

Answer 4

In 2D graphics, a vector is considered immune to the translation component in an affine coordinate transformation. An affine transformation can be decomposed into a translation vector (how much the origin is displaced) and a 2x2 linear transformation matrix (combination of scaling, rotating, and/or shearing effects). When a vector is transformed, it is not displaced by the translation because it has no 'place'; but it is subject to scaling, rotating and shearing.

Answer 5

When you first learned about vectors in high school, you probably weren't in Math class. You were very likely in your introductory Physics class. Were you to post this question in http://Physics.stackexchange.com you would get the answer: "Because a vector IS a representation of magnitude and direction."

We bend our tools to suit our problem domain. And in physics, the magnitude and direction model of a vector is the useful model.

Answer 6

Vectors can come up in very different contexts, and the only thing all things called "Vector" have in common is that they are all members of a vector space, that is, you can add them (and they form a group under addition), and you can multiply them with a member of some field (and there are certain laws for this multiplication).

However when you learn vectors in school, you typically learn first very specific vector spaces, namely the real vector spaces of two and three dimensions. And here the arrow representation is just more helpful than identifying vectors with points. To begin with, there's a clear, easily identified zero arrow of length $0$, but there's no natural "zero point". You can choose an origin, but that choice is arbitrary, and doesn't capture the special properties of the null vector very well (mathematically: The Euclidean space is best modelled by an affine space, while vector spaces are linear spaces). Moreover, vector addition is quite simple with arrows (you just chain the arrows). Note that for vector addition to work that way, you need the arrows to be freely movable. And the inverse (negative) vector is also obvious: You just reverse the direction. Now of course you could construct some pictures also for points. For example, the sum of two vector points is the fourth point of the parallelogram formed by the origin and the two vector points. However, just as the choice of origin is arbitrary, that addition rule also seems very arbitrary. As opposed to arrows, where the addition rule just seems natural (indeed, I guess even people who would never have heard about vector spaces, if asked "what would be a possible way to combine two arrows in order to get a third one, if you may move them, but not change their direction?" would get the idea to put one after the other; there's not much else you can do).

Moreover, think about where the word "vector" comes from. It comes from the Latin word "vehere", to carry. So its origin is intimately connected with moving objects around, that is, with translations; and indeed, the translations of the $n$-dimensional Euclidean space form an $n$-dimensional real vector space. And what's the most obvious representation of a (non-rotating) movement? Well, an arrow which goes from a starting point to the point where this starting point gets moved to. And if we translate the complete Euclidean space, it of course doesn't matter where we draw the arrow, all arrows of same length and direction describe the same translation.

Moreover I disagree that the coordinate numbers are more important than the direction. The coordinate numbers are dependent on the basis you choose. That is, give me an arbitrary non-null vector in an $n$-dimensional vector space, and an arbitrary list of $n$ numbers (which must come from the vector space's field, of course), and I'll give you a basis for which the vector is described by this set of numbers (and if you go for points in an Euclidean space, I'll even remove the restriction to non-null vectors, because I then just choose another point as the origin). On the other hand, the concept "two vectors have the same direction" is deeply ingrained into the vector space structure: Two vectors $v$, $w$ have the same direction iff they are linearly dependent (you might want to exclude the null vector in order to make the direction uniquely defined, and if your field is ordered, like the real numbers, you might add the condition that tthe factor between the vectors is positive, so that $v$ and $-v$ are not considered the same direction). Moreover, while comparing the lengths of arbitrary vectors needs some extra structure (namely the scalar product of two vectors), which of two vectors in the same direction is larger, and by how much, is well defined for any vector space with an ordered field (and if you drop the "which is larger", then it's integral, defining part of any vector space).

Note also that even for numbers, a more careful distinction between "numbers which describe points on the number line" and "numbers which describe differences on the number line" can sometimes be useful. For example, think of temperature scales. If you have a temperature in Fahrenheit and want to know it in Celsius, you have to use a different formula than if you have a temperature difference in Fahrenheit and want to know it in Celsius. So for this case, it is crucial that you know whether you are describing a point on the number line describing temperatures, or if you are describing a vector (difference between points).

Note that the same applies for space when doing transformations (translations, rotations, scalings); indeed, the Fahrenheit/Celsius example is just a special case of translation/scaling on the real line. Now if you look at the Euclidean plane, and you do e.g. a rotation around a given point (which need not be the origin, assuming you've chosen one), then whether and how a point changes depends on where this point sits; if you have two people who have chosen different origins and use vectors to describe points relative to the origin but use the same directions and lengths for their basis vectors, will find different rules to transform those location vectors. On the other hand, for "true" vectors they'll find the same transformation rules (remember, I assumed they used the same directions and lengths of basis vectors, only the origin is different). And for translations, you'll find that the location vectors change, while other vectors do not change at all.

Moreover the distinction between vector and point gets more important as soon as you leave the Euclidean space and go onto general manifolds. In that case, it is not possible at all to associate points and vectors; indeed, vectors are then even no longer defined all over the space, but are defined only in a certain point (that is, each point carries its own vector space). That is, the arrow picture gets even more appropriate, because then even the starting point of the arrow gets a significant meaning: The point to which the vector belongs.

The same is also true for the main (often only) field where you'll use vectors in school: Physics. A force is a vector, but it generally matters where the force is applied. The same magnitude and direction can have completely different, even opposite, effects if applied at different points of the same object (think e.g. of torsional moment).

So in short: * Arrows are superior to points for introducing vectors. * Vectors and points are in general not equivalent.

Maybe arrows are not the best way to introduce vector spaces, but they are definitely not the worst way.