Below are my attempted definitions of the two terms. Are these correct and do they clearly distinguish between the two terms?

My understanding is that the displacement vector comes first and that is then used to define the position vector.

Definition 1. The displacement vector between two points $A$ and $B$ is the vector $\overrightarrow{AB}$.

Definition 2. The position vector of a point $A$ is the displacement vector between the origin $O$ and the point $A$.

Another related question: If Definition 1 above is "correct", what then, if any, is the formal distinction between a displacement vector and a vector?

Related: 1, 2.


Fundamentally, they are all just vectors (you can apply the same math to them and the same axioms hold). The distinction in the names is merely a convenient way to convey their purpose more concretely.

You are correct that a displacement vector between $A$ and $B$ is generally the vector $\vec{AB}$ defining the displacement a particle would make going from point $A$ to point $B$.

The position vector is just the (displacement) vector from some arbitrary but defined reference point, usually called the origin, $O$.

The key to remember in the end however is that they are all just vectors ie: $vec{AB}$, $vec{OA}$... and that they all share the same properties and operations. There is nothing special about the position vector vs the displacement vector other than the name allowing us to quickly associate it with a given starting point.

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  • $\begingroup$ If my answer helped your understanding it would be terrific if you could mark it as "accepted". $\endgroup$ – fhorrobin Jul 13 '18 at 12:21

The formal notion you're looking for is that of an affine space. You have already linked a question (2) to which I've given a fairly thorough technical answer, so I'll avoid the formalism here.

Yes, you are essentially correct. Displacement vectors can be identified with the "difference" between two points, and are fundamentally baked-in to the structure of an affine space. If you then choose an origin point, you can define the position vector corresponding to any individual point as the displacement vector between that point and your chosen origin.

Therefore, displacement vectors are ever-so-slightly more fundamental in the sense that position vectors require you to make an additional choice of origin from the set of points in your affine space, while the displacements themselves require no such additional structure.

To answer your follow up question, the term "vector" shows up in a wide variety of distinct but related contexts. If you want a formal definition, then I would say that that a displacement vector is an element of a vector space which is incorporated into an affine space.

More generally, a vector is simply an element of a vector space. This is perhaps among the least satisfying answers imaginable, but there's really no other way to say it. The vectors which arise in differential geometry (i.e. tangent vectors to a manifold) are completely different beasts to the vectors which we've discussed here.

Lastly, in physics we often define a vector by how its components transform under rotations (or Lorentz transformations, in the case of 4-vectors). You can take that as the defining characteristic, but this can be overly broad$^\dagger$ and requires a careful discussion of symmetry transformations to treat properly.

$^\dagger$For instance, in SR the gamma matrices $\gamma^\mu$ transform like 4-vectors despite not actually being 4-vectors.

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