Distinction between vectors and points

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?

Solution 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.

Solution 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.

Solution 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.