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?
Best Answer
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.