TL;DR: I don’t like the way vectors are presented in calculus-based and algebra-based introductory physics. I think a more formal approach is warranted. This post addresses the problem of taking the magnitude of both sides of simple vector equations. If you want the details, read on.
This is the first post in a new series in which I will present a more formal approach to vectors in introductory physics. It will not have the same flavor as my recently begun series on angular quantities; that series serves a rather different purpose. However, there may be some slight overlap between the two series if it is appropriate.
I am also using this series to commit to paper (screen, really) some thoughts and ideas I have had for some time with the hope of turning them into papers for submission to The Physics Teacher. I’d appreciate any feedback on how useful this may be to the community.
To begin with, I want to address issues in the algebraic manipulation of vectors with an emphasis on coordinate-free methods. I feel that in current introductory physics courses, vectors are not exploited to their full potential. Instead of learning coordinate-free methods, students almost always learn to manipulate coordinate representations of vectors in orthonormal, Cartesian coordinate systems and I think that is unfortunate because it doesn’t always convey the physics in a powerful way. Physics is independent of one’s choice of coordinate system, and I think students should learn to manipulate vectors in a similar way.
Let’s begin by looking at a presumably simple vector equation:
The object is to solve for given . Don’t be fooled; it’s more difficult than it looks. In my experience, students invariably try to divide both sides by but of course this won’t work because vector division isn’t defined in Gibbsian vector analysis. Don’t let students get away with this if they try it! The reasons for not defining vector division will be the topic of a future post.
(UPDATE: Mathematics colleague Drew Lewis asked about solving this equation by combining like terms and factoring, leading to and then to . This is a perfectly valid way of solving the equation and it completely avoids the “division by a vector” issue. I want to call attention to that issue though, because when I show students this problem, they always (at least in my experience) try to solve it by dividing. Also, in future posts I will demonstrate how to solve other kinds of vector equations that must be solved by manipulating both dot products and cross products, each of which carries different geometric information, and I want to get students used to seeing and manipulating dot products. Thanks for asking Drew!)
One could simply say to “take the absolute value of both sides” like this:
but this is problematic for two reasons. First, it destroys the sign on the righthand side. Second, a vector doesn’t have an absolute value because it’s not a number. Vectors have magnitude, not absolute value, which is an entirely different concept from that of absolute value and warrants separate consideration and a separate symbol.
We need to do something to each side to turn it into a scalar because we can divide by a scalar. Let’s try taking the dot product of both sides with the same vector, , and proceed as follows:
This is a better way to proceed. It’s formal, and indeed even pedantic, but I dare say it’s the best way to go if one wants to include dot products. Of course in this simple example, one can see the solution by inspection, but my goals here are to get students to stop thinking about the concept of dividing by a vector and to manipulate vectors algebraically without referring to a coordinate system.
Let’s now look at another example with a different vector on each side of the equation.
Once again the object is to solve for given and . Note that solving for either or is obviously trivial so I won’t address it; it’s simply a matter of scalar division. Solving for is more challenging because we must again suppress the urge to divide by a vector. I will show two possible solutions. Make sure you understand what’s being done in each step.
We get two solutions, and they are geometrically opposite each other; that’s the physical implication of the signs. (I suppose we could argue over whether or not to just take the principal square root, but I don’t think we should do that here because it would throw away potentially useful geometric information.) We can find a cleaner solution that accounts for this. Consider the following solution which exploits the concepts of “factoring” a vector into a magnitude and a direction and the properties of the dot product.
See the geometry? It’s in the factor . If and are parallel, this factor is and if they are antiparallel it is . Convince yourself that those are the only two options in this case. (HINT: Show that each vector’s direction is a scalar multiple of the other vector’s direction.) This solution won’t work if the two vectors aren’t collinear. If we’re solving for then both vectors are assumed given and we know their relative geometry.
Let’s look at another example from first semester mechanics, Newton’s law of gravitation,
where and should be read as “the position of 1 relative to 2.” Let’s “take the magnitude of both sides” by first writing in terms of its magnitude and direction, dotting each side with a vector, and dividing both sides by the resulting common factor.
Okay, this isn’t an Earth-shattering result becuase we knew in advance it has to be the answer, but my point is how we formally went about getting this answer. More specifically, the point is how we went about it without dividing by a vector.
Let’s now consider a final example from introductory electromagnetic theory, and this was the example that got me thinking about this entire process of “taking the magnitude of both sides” about a year ago. It’s the expression for the electric force experienced by a charged particle in the presence of an electric field (obviously not its own electric field).
That one vector is a scalar multiple of another means the two must be collinear, so they must either be parallel or antiparallel. An issue here is that is a signed quantity. Again, we have a choice about which vector with which to dot both sides; we could use or we could use . If we use the former, we will eventually need to take the square root of the square of a signed quantity, which may lead us astray. Therefore, I suggest using the latter.
This may look overly complicated, but it’s quite logical, and it reflects goemetry. If is negative, then the dot product will also be negative and the entire quantity will be positive. If is positive, then the dot product will also be positive and again the entire quantity will be positive. Geometry rescues us again, as it should in physics. We can also rearrange this expression to solve for either or with the sign of properly accounted for by the dot product. By the way, and can’t be orthogonal becuase then their dot product would vanish and the above expression would blow up. Geometry and symmetry, particularly the latter, preclude this from happening.
In summary, “taking the magnitude of both sides” of a simple vector equation presents some challenges that are mitigated by exploiting geometry, something that is neglected in introductory calculus-based and algebra-based physics courses. I suggest we try to overcome this by showing students how to formally manipulate such equations. One advantage of doing this is students will see how vector algebra works in more detail than usual. Another advantage is that students will learn to exploit geometry in the absence of coordinate systems, which is one of the original purposes of using vectors after all.
Do you think this would make a good paper for The Physics Teacher? Feedback welcome!
You may not agree that the topic(s) of this question belong in an introductory calculus-based physics course, but I’m going to pretend they do for the duration of this post. Gradient, divergence, and curl are broached in Matter & Interactions within the context of electromagnetic fields. Actually, gradient appears in the mechanics portion of the course.
One problem with these three concepts, especially divergence and curl, is the distinction between their actual definitions and how they are calculuated. The former are rarely, if ever, seen at the introductory level and usually first appear in upper level courses. However, some authors [cite examples here] replace the physical definitions with the mathematical symbols invented by Heaviside and Gibbs to represent the calculation of these quantities. In other words, the divergence of is frequently defined as and the curl of is frequently defined as . These should be treated as nothing more than symbols representing their respective physical quantities and should not be taken as equations for calculation. If one insists on keeping this notation, then the dot and cross should at least be kept with the nabla symbol so that represents divergence and represents curl. Either way, these are operators that operate on vectors and their symbols should reflect that concept and should be interpreted as such and not as a recipe for calculation. This book by Tai was extremely helpful in getting this point across to me.
Gradient has its own unique problem in that some sources claim that one can only take the gradient of a scalar, which is patently false. One can indeed take the gradient of, for example, a gradient but the object one gets back is not a vector. If we adopt a unified approach to vector algebra and vector calculus we find that there are patterns associating the operand and the result when using these vector opators. For example, operating on a vector with doesn’t produce a vector; it produces a second rank tensor. This is one reason I would love to find a way to bring this approach into the introductory course. So many things would be unified.
But now, on to the questions I want to ask here
(a) Write a conceptual definition of gradient in words.
(b) Write a mathematical definition of gradient that does not depend on any particular coordinate system. You must not use the nabla symbol.
(c) Write a conceptual definition of divergence in words.
(d) Write a mathematical definition of divergence that does not depend on any particular coordinate system. You must not use the nabla symbol.
(e) Write a conceptual definition of curl in words.
(f) Write a mathematical definition of curl that does not depend on any particular coordinate system. You must not use the nabla symbol.
(Note: I need to revisit this post in the future to make sure the notion of applying gradient to a vector quantity can be handled in the coordinate free way I have in mind. My intuition is that it can be, but I need to work out some details. )
In this post, I will address the first question on the list in the previous post. What exactly does it mean for something to be a vector?
In almost every introductory physics course, vectors are introduced as “quantities having magnitude and direction” and are eventually equated to graphical arrows. A vector is neither of these, but is something far more sophisticated. Remember that I’m coming at this as a physicist, not a pure mathematician. I will probably get more than a few things incorrect. Let me know if/when that happens. Let me see if I can present this at a level suitable for an introductory calculus-based physics course. Imagine you walk into class on the first day and start talking. Here goes.
We live in a Universe with has measureable properties, and containing physical entities that also have measureable properties. A lot of physics consists of attempting to measure, and thus quantify, these properties (experiment). More important to some physicists is describing these properties mathematically and making predictions about them (theory) rather than attempting to measure them. We can invent mathematical objects to represent these measureable properties. The word represent is important here, because the mathematical object representing an entity is not the same thing as the entity itself. These mathematical objects themselves have properties, and these properties allows us to manipulate these objects so as to use them to make predictions about Nature.
The properties possesed by the mathematical objects we use to describe Nature collectively form something with a very strange name: a vector space. That sounds very technical and complicated. It is indeed a very technical term because it means something profound. However, as I will try to convince you now, it is not necessarily complicated at all. Let me attempt to show you.
I will use bold symbols (e.g. etc.) to represent mathematical objects with the properties that collectively form a vector space. These mathematical objects have a generic name: vectors. Yes, that’s their name. Note that there is nothing at all here to do with arrows or anything else really. Vectors are nothing more than mathematical objects with properties that let us model and make predictions about the properties of the Universe we observe and try to understand in Nature. Be careful to understand that there are two sets of properties here, those of the Universe and its inhabitant entities, and those of the mathematical objects we use to represent those things. I’m not saying this is the best way to describe this, but it’s a start.
I will use italic symbols (e.g. etc.) to represent ordinary numbers you are already familiar with. Technically, the are real numbers and every math course you have ever taken has used them whether or not you knew they had a name.
In a vector space, there are two and only two mathematical operations defined: addition and scalar multiplication. That’s all there is. You’ve known how to add and multiply for a long time, and there is nothing new to see here. Consider addition. So, in any vector space, you can take any two vectors, and add them and get a third vector. It’s just that simple, however, I must warn you that there is indeed something deeper going on here but there’s no need to bring it up yet because it’s a geometry issue. We will get to it soon enough. So for now, addition is the same addition you’ve already become familiar with. Oh, here’s a new technical gem for you. The simple fact that adding two vectors gives you a third vector is a property that we use to say that the vector space is closed under addition. All that means is that when you add two vectors, you get a vector. All three inhabit the vector space. That’s simple to understand. You cant add two vectors and get, say, a real number. You must always get a vector. That’s very simple. Let’s say it more mathematically.
- In a vector space, addition is a closed operation. If and are vectors then is also a vector.
Now consider scalar multiplication. You’ve known how to multiply real numbers for a long time, and again, there isn’t much new to see here. Multiplying a scalar and a vector gives another vector. We will explore the goemetric implication of this later. Like vector addition, scalar multiplication is a closed operation.
- In a vector space, scalar multiplication is a closed operation. If is a vector and is a scalar, then is also a vector.
Here is a list of remaining properties that define a vector space.
- In a vector space, addition is commutative, meaning that the order of the vectors being added doesn’t matter.
- In a vector space, addition is associative, meaning vectors can be grouped in any way as long as the order isn’t changed.
- In a vector space, scalar multiplication is associative. The things you’re multiplying can be grouped differently as long as their order isn’t changed.You get the same vector either way. Cool!
- In a vector space, when you have the sum of two scalars multiplying a vector, the thing you get back is the sum of each scalar multiplying that vector.
- In a vector space, scalar multiplication is distributive over vector addition. Some authors equivalently say that vector addition is linear. Both of these mean the same thing, but I think the second way of saying it is more important, and I will try to show why later. When you have a scalar multiplying the sum of two vectors, the vector you get back is the sum of that scalar multiplying each vector separately.
- In a vector space, there is a multiplicative identity element such that multiplying it by any vector you get the same vector back. This effectively defines a unity element, commonly called 1 (one). This is important because sometimes we can exploit what I like to call a “sneaky 1” to help manipulate a mathematical expression. More on that when we need it.
- In a vector space, there is an additive identity element such that adding it to any vector gives that same vector back as the sum. This is effectively a definition of a zero vector.Seeing zero written this way (as a vector) may seem strange, but you will get used to it.
- In a vector space, there is a member of the vector space called an inverse element such that adding it to any vector gives the identity element (zero element). For any vector we have a vector such that the two sum to zero. Do not think of the sign as subtraction. Think of it as merely a symbol that turns the vector in to its additive inverse.
We’re done. That’s it. These properties collectively and operationally define a vector space that is inhabited by mathematical objects called vectors. These properties also define the things we can do to manipulate vectors. Note there is no mention of subtraction, and there is no mention of division. There is vector addition and scalar multiplication. That’s all there is. This is really simple! Also note there is no mention of magnitude, direction, arrows, components, dot products, or cross products. If you don’t know what those three terms mean don’t worry. We will define them later.
Let me now convince you that you have dealt with vector spaces and vectors for many years and didn’t realize it. Consider the real numbers (that’s all positive numbers, negative numbers, and zero regardless of whether they’re rational or not, and regardless of whether they’re integers or not). Do they meet each and every one of the properties above? To convince yourself that they do, go through them one by one. Does adding two real numbers give a real number? Yes (3.2 + 5.9 = 9.1). Does adding 0 to 5 give 5? Yes. Does adding 6 to -6 give 0? Yes. You can do the rest. Therefore, I claim that without knowing it, you have been using vector spaces and vectors all along!
Now, let me ask you a new question. Consider only the natural numbers. Recall that these numbers are the ones you use for counting and you’ve probably been using them longer than you’ve been using real numbers! Do the natural numbers (counting numbers) form a vector space with each number being a vector? I will tell you that the answer is no, they do not, but I don’t want you to take my word for it. Go through each of the above properties one by one using counting numbers and see if you can convince yourself that these number do not inhabit a vector space.
This is a physics class, so let’s get more physicsy. In physics, as in all science, we use a system of units called the SI System. All scientists know about this system of units, but some subdisciplines (e.g. astrophysics) don’t use them yet. I hope this changes because it will make many things simpler, but I digress. The SI System consists of seven independent fundamental units that represent seven fundamental quantities: mass, length (I prefer spatial displacement), time (I prefer temporal displacement), thermodynamic temperature, amount, luminous intensity, and electric current. All physically measureable properties in our Universe can be expressed in various combinations of these seven fundamental quantities and their units. Your question is: Do these seven fundamental form a vector space? What a weird question! Still, it’s one you can address by, again, working your way through the defining properties of a vector space given above. See what you can come up with.
This may seem a very strange way to begin introductory physics, and it is! It’s strange, but I hope it will help get you to a place where your understanding is deeper than it would be had we begun in a traditional way. Accept the strangeness and uncomfortableness you feel right now, and then let it go. There’s much learning to be done, and it starts here.
This is the first in a series of posts in which I want to share some hopefully interesting things about mathematical descriptions of rotational motion. This series was inspired by a talk given at the 2015 winter AAPT meeting in San Diego. The author claimed to have found a way to represent angular displacement as a vector (true, such an expression exists and is not widely used) and that angular displacements commute (false, in general they do not except when infinitesimal). The same author presented an updated poster on this topic at the recent winter meeting in Atlanta. In researching the arguments presented in these two talks, following up on the references therein, and in searching the undergraduate and graduate physics and mathematics teaching literature on descriptions of angular quantities, I stumbled onto some of the most interesting topics I’ve ever encountered. As you may have already guessed, I want to find ways of bringing these gems of understanding into the introductory courses so students won’t be so mystified when then encounter the in upper level courses. By the way, the papers from these talks aren’t availble online; I only have paper copies and I do not have the author’s permission to distribute them.
I am sure most of this will be trivial for many readers, so apoligies in advance. Even though I too studied out of Goldstein in grad school, it was not the case that all my existing conceptual mysteries were solved. As always, I tend to frame things from the point of view of that introductory physics student for whom we want to provide an unparalleled physics experience. I don’t want that student to ever say, “Well that was never pointed out to me in intro physics.” I want that student’s conceptual foundation to be better than mine was when I was that student.
In this initial post, I will list as many of the questions I can think of that arose as I researched this topic. I will not answer any of them in this post, but will attempt to do that in subsequent posts. I will put the questions into some preliminary order, but I can’t guarantee that order won’t change later. Some questions may change to more accurately reflect what I’m trying to explain.
- What does it mean to be a vector?
- What do vector dot products and vector cross products mean geometrically?
- What is the physical significance of the double cross product (aka triple cross product)?
- Is there a coordinate free expression for the total time derivative of a vector?
- Is there a coordinate free expression for the time derivative of a unit vector (a direction)?
- Can angular velocity be described as a vector?
- Can angular displacement be described as a vector?
- If work is calculated as the dot product of two vectors, then when calculating rotational work how can angular displacement not be a vector?
- If angular velocity is a vector, shouldn’t its integral also be a vector and not a scalar?
- Why does translational displacement commute?
- How, if at all, are translation and rotation (revolution?) related?
- Why do infinitesimal angular displacements commute?
- Why do finite angular displacements not commute?
- What is the distinction between rotation and orientation?
- Is angular velocity the derivative of a rotation?
- So then what is angular velocity the derivative of anyway?
- Can angular velocity be integrated to get angular displacement?
- Can these ideas be brought into the introductory calculus-based or algebra-based physics courses?
I think that’s all, at least for now. I don’t claim this list to be comprehensive. The number of questions isn’t significant either. Let’s see where this goes.
(a) In first semester physics, you learned that mass is necessary to calculate the gravitational force shared by two interacting entities. What are the physical implications of mass always being positive?
(b) In second semester physics, you just learned that charge is necessary to calculate the electric force shared by two interacting entities. What are the physical implications of charge bring both positive and negative?
This question might serve as a final exam for an introductory physics course. It could serve that purpose for my own courses, but it may not be appropriate for your courses so don’t worry if that’s the case. If you do not include system schemas in your course then this question won’t make any sense to you. I recommend this paper by Lou Turner in The Physics Teacher for familiarizing yourself with system schemas.
(a) Consider four entities in the same region of space, each of which interacts with the other three through some interation, the exact nature of which is not important for our purposes. Draw a schema showing four entities and all of their mutual interactions. Label the entities as 1, 2, 3, and 4.
(b) Draw a system boundary around the system consisting of entities 2 and 4. Explicitly label this sytem.
(c) For this system, write the momentum principle, the energy principle, and the angular momentum principle as specifically as you can given what you know about the system and the four entities. You may feel free to make any necessary (thermodynamic) assumptions for certain approximations to be made as long as you explicitly state them.
(d) For this same system, identify any interactions that could be accounted for using interaction energy (a scalar) rather than force (a vector).
(e) Now, draw a system boundary around the system consisting of all four entities. Explicitly label this system.
(f) For this system, write the momentum principle, the energy principle, and the angular momentum principle as specifically as you can given what you know about the system and the four entities. You may feel free to make any necessary (thermodynamic) assumptions for certain approximations to be made as long as you explicitly state them.
(g) For this same system, identify any interactions that could be accounted for using interaction energy (a scalar) rather than force (a vector).
(h) For the two systems defined above, what would make one preferable over the other? Be as specific and as comprehensive as you can be.
(i) Assume nonrelativistic circumstances in both systems. What, if anything, would change in either schema if you were to transform into a new reference frame?
I will probably add to this list of parts as I think of good items to add. Feedback welcome!