Distinction between vectors and points

Question

I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ for points and $[x,y,z]$ for vectors. We always had to translate the point $P=(a,b,c)$ to the vector $\overrightarrow{OP} =[a,b,c]$ before we started calculating with them.

Now, after I started at the university, people don't seem to care anymore. My professors either say that they're the same, or that they're almost the same, and the books I have seem to share that view. The book I use for my calculus course (Colley's Vector Calculus) says, among other things, the following:

[...] we adopt the point of view that a vector field assigns to each point $\textbf{x}$ in X a vector $\textbf{F}(\textbf{x})$ in $\mathbb{R}^n$, represented by an arrow whose tail is at the point $\textbf{x}$.

So it seems like a point is also a vector.

My question is this: Do mathematicians distinguish between points and vectors, and if they do, in what circumstances?

Answer 1

A point in Euclidean space is properly regarded as an element of an affine space rather than a vector space. That's because vector spaces have a distinguished origin, and "space" in the general sense doesn't: you can move the origin anywhere you want. Affine spaces are also constructed to have the property that the difference between two points is a vector. Because affine spaces don't have a distinguished origin, you can't add two points in an affine space, but you can take affine combinations.

There is also a more general notion of "point" as just an element of any set equipped with some kind of geometric structure, such as a point in a topological space.

Answer 2

In general, mathematicians would distinguish between points and vectors in a context where that distinction is important, and might not bother to distinguish between them in a context where it isn't important.

Answer 3

I would say it's a good habit to distinguish points from vectors (in the context that I think you're referring to), even at university!

Geometrically, any point looks just the same as any other point, whereas not all vectors are equal; two vectors can have different lengths, for example, and there is one very special vector which has length zero. And to talk about the coordinates of a vector, what you need is only a basis, but to talk about the coordinates of a point you need a basis and an origin (an arbitrarily selected reference point).

However, converting points $P$ to vectors $\overrightarrow{OP}$ is strictly speaking not necessary (and in my opinion a bit artificial actually). You can instead use the geometrically natural operations "point + vector = point" and "point – point = vector" (but, as Qiaochu already said, not "point + point", which is geometrically meaningless). The textbooks insist on using the vector $\overrightarrow{OP}$ just so that they can express things in terms of the operation "vector + vector = vector" and don't have to introduce those other operations.

Answer 4

It's a good habit to distinguish the coordinates of a point from vectors. As everyone else has pointed out, Euclidean space is special — but I'll add that on top of that, cartesian coordinates on Euclidean space is special. If you use, for example, polar coordinates on Euclidean space, you'll find that you can't subtract the coordinate components of different points to obtain the components of the displacement vector. For instance, the displacement vector from the point $(r, \theta)$ to the origin is $r \mathbf{e}_r$, not $r \mathbf{e}_r + \theta \mathbf{e}_\theta$, where $\mathbf{e}_r$ and $\mathbf{e}_\theta$ are the normalised coordinate basis. It only gets worse when you start working with general curved spaces.

(A pet peeve of mine is when people talk about position vectors, especially in the context of non-affine space.)

Answer 5

The simplest way to understand the importance of distinguishing points from vector is to consider subspaces. For example, imagine a plane $P$ in $\mathbb{R}^3$ that does not contain the origin. Then, if you add coordinate-wise two points of $P$, the result is not in $P$. The operation point minus point gives as a result a vector in $\vec P$, the direction of $P$, namely the plane parallel to $P$ and that contains the origin. Now the addition of vectors in $\vec P$ stays into $\vec P$, and the addition of a vector (in $\vec P$) and a point (in $P$) gives a point (in $P$).

In a differential geometric setting, vector are all based at a given point. So if you take this point of view, a vector in $\mathbb{R}^n$ should bedefined by $2n$ coordinates. This amount to forget that you can compare (i.e. define equality of) vector based at different points.