I have a few (possibly very stupid) questions relating to position vectors; more specifically my confusion about them.

Following Halliday and Resnick's text, we define vectors by their magnitude and direction, but not by their 'location' in space. They give as an example the displacement vector, and draw three of the same vector in different locations to emphasize that shifting the vector does not change it.

Then in the next chapter we are introduced to position vectors relative to a given origin, which is a vector extending from the origin to the position of the particle.

How are we to think about position vectors? It seems like location is also important, even though we only defined vectors as having magnitude and direction.

How do we think about the addition of a displacement to a position vector? We define displacement vectors as the difference between two position vectors. Mathematically this is the same as saying that the final position vector is the result of adding the displacement vector to the initial position vector. The vector algebra doesn't care which ones are assigned the label of 'position' or 'displacement'; do we simply agree that when we add a 'displacement' vector to a 'position' vector, we get a 'position' vector?



This is one of those things that (intentionally) gets conflated, though it may be better if we were more consistent about keeping them separate.

So, points don't form a vector space. It makes no sense to ask "what's the location of New York plus the location of DC". However, given two points we can subtract them and get a displacement, and we can add that displacement to points to get new points. The mathematical structure for this is called (among other things) a torsor.

Your text is, however, accurate. If we choose a particular point to be our origin, call it $O$, then we can make a vector $r = P - O$ and call it a position vector for $P$. Now the difference between a position vector and a "regular" vector is that it changes when we change what we consider the origin. When we perform a translation on a system, position vectors change, regular vectors do not.

  • $\begingroup$ you're right about points not belonging to a vector space. However, they belong to an affine space, with displacements at a point forming a vector space, and a result displacements can be added to position vectors because vector addition to an element belonging to a affine space is allowed. $\endgroup$
    – Bruce Lee
    Feb 8 '16 at 5:11
  • 1
    $\begingroup$ @BruceLee An affine space is a torsor. $\endgroup$ Feb 8 '16 at 5:17
  • $\begingroup$ I'm going to read that link on torsors more carefully, but if we have a framework to handle the addition of points and displacements to get new points, why do define a position vector? $\endgroup$
    – 428
    Feb 8 '16 at 5:23
  • $\begingroup$ @428 Mainly convenience, and it usually doesn't cause any problems (but when it does, they can be sticky.) Wait 'til you get to "axial vectors". *sigh* $\endgroup$ Feb 8 '16 at 5:31

A vector is a basic mathematical construct. There can be many types of vectors (velocity, position, force, etc...). Your first chapter is defining a vector in general. The second is introducing a vector that describes position. If you take the difference of two position vectors, it is a displacement vector. Thus. Through algebra the difference of a position with a displacement is a position.

This is a bit of abuse of the word position from chapter 1 to 2. I think it would be more clear in chapter 1 if they say that vector algebra works irrespective of your coordinate system.


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