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I have noticed working with bright undergraduates that it is not uncommon for them to have difficulty easily converting between a point—say, a point $p$ on a surface $S \subset \mathbb{R}^3$—and a vector $v$ from the origin $o$ of the coordinate system in which $S$ is embedded: $v=p-o$. For example, a normal vector $n$ at $p$ might be computed as based on $o$, and then needs to be moved out to $p$ for display on the surface of $S$: $n + v$. To me the transition between points and vectors is natural and easy, but not so to my students.

Having learned from @AlexandreEremenko's answer to my question, When were vectors invented?, I now see that however natural vectors are to us now, they were not natural in the past. And perhaps this is a hint that there is an intellectual hurdle here I am not seeing.

I'd appreciate comments on this issue. Wherein lies the difficulty converting between points and vectors out to those points?

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  • $\begingroup$ What is the context? A sophomore course in multivariate calculus? $\endgroup$
    – user507
    Commented Apr 18, 2019 at 16:05
  • $\begingroup$ @BenCrowell: Context: Graphics. One often needs to design an object around the origin, but then place that object out in the "world" by adding a translation vector to each point. $\endgroup$
    – Joseph O'Rourke
    Commented Apr 18, 2019 at 17:27
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    $\begingroup$ This is not an answer, but the question makes me think about the Gauss map from an oriented surface $X$ in ${\bf R^3}$ to the unit sphere $S^2$. For $p\in X$, the Gauss map takes $p$ to the point on the unit sphere corresponding to the unit normal vector at $p$. This is a simple, intriguing construction which for a little while, challenges math majors in a differential geometry course. $\endgroup$
    – user52817
    Commented Apr 19, 2019 at 14:36
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    $\begingroup$ @user52817: I like that example. I use the Gauss map regularly.Thanks. $\endgroup$
    – Joseph O'Rourke
    Commented Apr 19, 2019 at 14:55

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I strongly dislike the “magnitude and direction” description. Yes, it comes from physics, and is helpful sometimes, but it hides the true nature of vectors, and makes later understanding more difficult (what is the direction of the zero vector?)

I prefer to focus from the very beginning on the core property of vectors: they can be added and rescaled; and more importantly, these two operations behave in a friendly manner with each other: scalar factors can be distributed among two added vectors.

In class I start with something along the line of “you cannot add your address plus mine, but you can add motion instructions. The address is akin to a point coordinate, the instruction can be applied anywhere that you are standing.”

Then I stand in the middle of the room, and ask the entire class to point to my nose: every arm points ‘differently’. After this, ask them to point North: every arm points ‘the same’.

Usually, these three approaches work together to give students a good grasp. To clinch it, I am meticulous about writing point coordinates like so (2,3), and vector coordinates like so <2,3>.

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    $\begingroup$ I like the point-north example. Although I can imagine a bright student realizing that pointing north from NYC is not the same as pointing north from Moscow. $\endgroup$
    – Joseph O'Rourke
    Commented Apr 19, 2019 at 10:39
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    $\begingroup$ Re the "magnitude and direction" description, that isn't actually how working physicists define vectors. We define them in terms of their transformation properties under rotations and parity transformations. The magnitude and direction idea is a common freshman-physics-textbook definition, not a physicist's definition. $\endgroup$
    – user507
    Commented Apr 19, 2019 at 15:30
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    $\begingroup$ @Ben Crowell Yes, I know what you mean. But a) as you mention it is an idea found in physics textbooks, so “coming from physics” as I said: and b) my issue is that many students arrive in a course already infected by that problematic interpretation. $\endgroup$
    – Rodrigo A. Pérez
    Commented May 4, 2019 at 10:52
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My first thought (which is just barely this side of a comment) is that when students are introduced to both vectors and points, they are always with respect to a specific origin. Even if lip service is paid to vectors as a magnitude and direction in some coordinate-free way, realistically the examples have the origin as "the origin".

So my conclusion is that at least part of this is that vectors are introduced this way, so people get used to it. This is an unscientific answer, though, so I welcome others - doubtless it is indeed more challenging to think of vectors in this more abstract way.

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You might want to look at how vectors are introduced to your students in previous coursework. At my school (a community college in California), it would be something like this:

  • The remedial trig course includes vectors in the course outline, but not all students take that course, and if they do, the instructor will often skip the topic.

  • A first semester physics course spends a lot of time on vectors. Probably some but not all of your students have taken such a course.

  • The sophomore multivariate calculus course introduces vectors from scratch on the assumption that students have never seen them before.

  • The linear algebra course spends most of the semester teaching students algorithms for manipulating matrices and row and column vectors. I doubt that students connect these manipulations in any way to any possible geometric interpretation.

If you ask your students to define what a vector is, you will probably get a variety of answers, depending on which of these experiences they've had. My guess it that some will say it's a thing that has both a magnitude and a direction (a common definition in physics textbooks), while others will say that it's an n-tuple of numbers (basically the only representation they ever seem to work with in the math courses). Neither of these is quite the same as the definition that would apply best to your example, which is that a vector is an equivalence class of displacements between points in space.

When I teach my students about vectors in freshman physics, I find that they need a lot of encouragement to think about vectors as "portable." For example, they may draw a free-body diagram in which two force vectors act on an object, and the force vectors are drawn tail to tail. Then if they want to sketch graphical addition of the vectors, they have to slide them around into tip-to-tail position. This is not natural to them, nor is it natural to them to understand that the vectors have to be slid around without rotation (parallel transported, in fancy terminology).

In my office hours, I spend a lot of time demonstrating this with pens laid on the tabletop. I place two pens on the tabletop, tail to tail, and ask the student if they can slide them around so they're tip to tail. Usually they don't understand how to do this. Then I usually take one pen, move it around some path and bring it back to the start again, to illustrate the idea of sliding without twisting. I may also explicitly demonstrate twisting, in order to show that it's not allowed.

In general, students usually do not get much practice with the conceptual aspects of vectors. For example, I have them do this think-pair-share exercise:

In example 4 [computing the components of a vector given in magnitude-direction form], we dealt with components that were negative. Does it make sense to classify vectors as positive and negative?

This may seem extremely elementary to a mathematician, but for many students it's very difficult.

For the particular example you give, of converting between points and vectors, note that for students who have learned about vectors in a physics class, their stock of mental examples of vectors probably has a couple of primary examples in it: force vectors and displacement vectors. For force vectors, it isn't even true that you can convert back and forth between a point form and a vector form, if the point is to be a point in the ordinary Euclidean space that we swim around in. It could of course be a point in a more abstract "space" of force vectors, but this is not a level of abstraction that would occur to students at this level.

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    $\begingroup$ 'think about vectors as "portable." ' I like this word in this context. $\endgroup$
    – Joseph O'Rourke
    Commented Apr 18, 2019 at 17:28

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