I've read Apostol's Calculus, in the section on analytic geometry. He says that he's going to use 'vector' and 'point' interchangeably.

But in Beardon's Algebra and Geometry, he argues that there is not agreement on what vectors are, some say that they are points on $\mathbb{R}^n$, some say that they are directed line segments and for others they are classes of line segments in which they are the same if they represent the same displacement.

I'm curious if problems can appear by assuming that each of the definitions given by Beardon is equivalent.

  • $\begingroup$ The collection of points - Euclidean spaces, and vector spaces are both abstractions of the same underlying object, namely $\mathbb R^n$. There certainly can be situations in mathematics where mixing up the concrete realizations of distinct abstract objects can run you into trouble, but I've never seen an instance where this particular mix-up causes problems. I would also be curious to see whether problems can occur here, and if someone thinks that they can't, is there a "rigorous" argument one can make to justify that no issues occur? $\endgroup$ – Dustan Levenstein Apr 30 '15 at 14:15

The point — pun intended — here is that one can think of points in Euclidean space as locations in space, without any reference to an origin or a coordinate system. As such, given two points $P$ and $Q$, the difference $Q-P$ (which we can visualize as the vector $\overrightarrow{PQ}$) gives a well-defined vector quantity, independent of choice of origin in our Euclidean space, whereas the sum of $P$ and $Q$ has no intrinsic meaning. However, if you persist, the sum of a point $P$ and a vector $\overrightarrow{PQ}$ does give a well-defined point. :)

The geometry of Euclidean space with no distinguished origin is often called affine geometry.

All that said, I don't usually make the distinction: When I teach multivariable calculus and linear algebra, we work in $\Bbb R^n$ as a vector space and conflate points and vectors.


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