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I'm trying to give a student a (physically) intuitive, heuristic explanation as to why certain quantities are vectors and others are scalars. Here is what I have come up with:

Scalars are quantities that are fully specified by their magnitude. Their value at a point has no direction associated with it and as such is invariant under rotations of coordinate systems (the value of a scalar at a point is independent of where the point is located relative to a particular coordinate system).

Vectors are quantities that require both a magnitude and a direction to fully specify them. Accordingly, a vector (in general) requires more than one number to fully specify it at a point $^{(\ast)}$; these numbers (the components of the vector at that point) are dependent on the coordinate system in which the vector is described relative to.

[$^{(\ast)}$ It is important to note that these numbers are specified relative to a particular coordinate system, thus the components of a vector are very much coordinate dependent.]

To clarify these definitions, consider an example for each:-

Vector example:

Position - this is a vector quantity as one must specify both a magnitude (the distance of the object relative to you) and a direction (whether it's North, South, etc... relative to you). If one only specified a distance, then this would not fully determine the position of the object relative to you as distance is independent of direction.

Scalar example:

Temperature - this is a scalar quantity since it only has a magnitude. For example, if you are in room, the temperature at the point you are situated in the room does not depend the direction that you are facing in the room - the temperature at that point will not change if you were facing North at that point and then turned around to face South.

I would very much appreciate feedback on what I've written. Is it a valid description?

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  • $\begingroup$ I see no flaws, you might also want to add that the distance is also a scalar in the vector example. $\endgroup$ – user83548 Dec 7 '15 at 15:42
  • $\begingroup$ This sounds great so far. Two suggestions: first, perhaps a couple more examples; second, watch out for the difference between a reference coordinate system (e.g. NorthSEW) versus a reference point (e.g. zero-temperature). $\endgroup$ – DilithiumMatrix Dec 7 '15 at 15:48
  • $\begingroup$ Easy: Ask him if he only cares about the speed of a bullet, or about the direction it's taking (with respect to his location). $\endgroup$ – Carl Witthoft Dec 7 '15 at 16:08
  • $\begingroup$ @DilithiumMatrix Thanks for the feedback. What would you suggest as a description to clarify the distinction between the two? Would it be something like, direction of a vector is an intrinsic concept - the number(s) used to describe it is (are) coordinate dependent, but it has a coordinate independent notion of direction (it describes the relative position of one point relative to another, independently of distance), whereas a reference point is an artefact of how a particular coordinate system is defined?! $\endgroup$ – Will Dec 7 '15 at 16:14
  • $\begingroup$ See this post. physics.stackexchange.com/q/191016/37364 $\endgroup$ – mmesser314 Dec 7 '15 at 16:18
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Personally, I think it's better to phrase discussions of mathematical objects as them being, 'tools to describe things' --- avoiding descriptions which suggest intrinsic properties that are instead associations. For example, a vector is not something with a magnitude and direction --- a vector is simply a set of numbers. A vector is very useful for describing something like a position or velocity, which also requires a set of numbers to describe, for example magnitude and direction $(r, \theta)$ or coordinate positions $(x,y)$. In my opinion, this also greatly demystifies the concept of a vector (or tensor): it's nothing magical, just an extension of a scalar.

As I mentioned in the comments, this meshes better with both types of objects, for physical purposes, requiring a reference. For scalars this is a zero-point (like zero-temperature), and for vectors this is a coordinate system (like Greenwich, and the cardinal directions).

Scalars and vectors both require reference systems. A scalar requires a single axis---imagine an $x$-axis, while a vector (e.g. 2D) requires two---an $x$- and $y$-axis.

In both cases the values can be transformed (translations and dilations) by changing the reference frame (or scaling), but for dimensions $n > 1$ there is the additional concept of a rotation.

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  • $\begingroup$ DilithiumMatrix Nice answer. The only problem I have is that, knowing what has happened in the past, the student will probably question (despite this description) what distinguishes a reference point for a scalar (such as 0C in the Centigrade scale) and the notion of something having a direction. Would it be best to elaborate on what you've written by saying that a scalar quantity similar requires a reference point to quantify it relative to in terms of a particular unit of measurement... $\endgroup$ – Will Dec 7 '15 at 16:38
  • $\begingroup$ ... whereas a vector quantity requires the introduction of a coordinate system to describe its components relative to (its direction quantified by the value of its component relative to each reference axis)?! $\endgroup$ – Will Dec 7 '15 at 16:40
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    $\begingroup$ I mean, saying a vector is "a set of two numbers" and not "something with magnitude and direction" suggests there's no isomorphism between "pairs of numbers" and "magnitude and direction". But there is such an isomorphism, essentially a change in coordinates ("$r$" is magnitude, "$\theta$" is direction). I think the problem here might be the word "direction", which is only just barely independent of the choice of coordinate system ("that a-way!") $\endgroup$ – levitopher Dec 7 '15 at 19:17
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    $\begingroup$ @DilithiumMatrix: I mean, I'm not even sure I want to try to determine if you could make a vector space out of (Temperature, Pressure). For starters, the units are not the same. Even if you scaled that out, you would have to imply something about an inner product which would be unphysical. There is a clear mathematical difference between "A Cartesian Product" and "A vector space", and mixing them is asking for trouble, even conceptually. $\endgroup$ – levitopher Dec 8 '15 at 2:46
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    $\begingroup$ ...further, the vector space (T,P) does not pass the "abstraction-experiment" - you won't get any answers wrong by making them individual scalars rather then just grouping them together arbitrarily. In fact, you will have additional pieces of information which are not useful ("length"). You are adding structure which is not necessary. $\endgroup$ – levitopher Dec 8 '15 at 2:48
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Your current explanation is essentially exactly what I usually tell students. The only additional piece which I would talk about (if I had time) would be the relationship to abstraction and experimental verification. I think of examples like "what about two cars hitting each other? The directions matter", but a student might say "how do you know the directions matter? This is a sheet of paper and I'm just doing it this way because you told me to!"

So then you need to take them through the abstraction-verification process (in this example you need to assume conservation of momentum is true. Not ideal, but at the moment this is what I can think of!):

1) Construct an abstraction of reality in which momentum does not have direction. Consider two cars of equal mass and equal speed colliding head-on and sticking together. Do the experiment, we see that cars are at rest afterwards. We can use conservation of momentum ("without direction") if we use a negative sign for one of the two momenta.

2) Now do the same experiment with two cars colliding at a 90 degree angle. The students will immediately see the direction post-collision, but they might not be convinced about why the final momentum isn't simply $2mv$ (or even 0). The best thing to do is actually perform the experiment (hockey pucks or whatever) to demonstrate that their abstraction does not reproduce the correct result.

3) Present the abstraction with momentum as a vector. Derive a prediction for the 90 degree collision, and use the experiment to verify the result.

Of course, depending on the context the experimental verification may simply have to be told to them, which is also not ideal. But in any case, students might recognize the basic hypothesis-testing argument, and might hopefully be convinced that some quantities need to be vectors!

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Actually the physical quantity speed is always a positive value. The person driving a car reads their speed from a speedometer which is always positive, when measuring average speed we dont worry about negative distances, when calculating average speed. In a sense when using the number line, we are using a geometric model to "vectorise" the distance and turn it into a displacement, positive to the right and negative to the left. Physics always has this problem of using abstract thematically ideas to represent real world physical systems. Look at how imaginary numbers are used in AC circuit theory, in a sense real numbers are no more real than imaginary numbers. The i or j makes solving equations that much easier in circuit theory.

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