This week, we transitioned to chapter 1 of the Matter & Interactions textbook (fourth edition). I have WebAssign problem sets for each chapter available for formative assessment and practice while working their way through the reading. I encouraged them to use the book the way it was intended to be used, specifically by stopping and doing the checkpoints in situ and NOT going forward until they can get the correct answers by working them out (checkpoint answers are provided at the end of every chapter). My expectation is for them to have worked their way through chapter 1 by Monday of next week.
In class, I spent two days introducing vectors. I have never been happy with the introductory course’s treatment of vectors, mainly because of inconsistencies in terminology and notation (sometimes within a given textbook). More importantly, vectors are almost always presented in terms of their coordinate representations and not as the inherent geometric entities they are, and this completely undermines the main reason vectors are used in the first place: to describe physics without the need for coordinates. If we want students to understand that physics is independent of both reference frame and coordinate system, then why not present it that way from the beginning? I think it might pave the way to tensors, which are nothing but an extension of the vector concept to slightly more complicated quantities.
So, I began by introducing a vector as a quantity which can be represented with an arrow. The arrow’s length encodes the quantity’s magnitude (i.e. how much? or how many?) while the arrow’s direction encodes the quantity’s direction. I don’t like saying that a vector is “a quantity with magnitude and direction” because there are quantities that have magnitude and direction but are not vectors (e.g. finite angular displacements). Equally troublesome is saying that a vector is “a list of three numbers that transform a certain way from one frame to another” because, well, HUH? Defining a vector as an “element of a vector space” is rare in physics contexts (I don’t think I’ve seen it done that way, at least not in the intro course) and it’s really no less circular than speaking of mysterious transformation properties. Saying a vector is “something which can be represented with an arrow” seemed a relatively decent compromise. I don’t know…maybe I could do better.
As I introduced these ideas, I “invented” symbols on the board for them and gave the appropriate LaTeX commands (defined in my mandi package) to get these symbols. I want students to start associating LaTeX with material from the textbook so we can be consistent with notation from the beginning. They know to use \vect{} to indicate the symbol for a vector quantity, \magvect{} (think “magnitude of vector”) to get the symbol for a vector’s magnitude (always with the appropriate unit…mandi knows every quantity’s SI unit in as many as three different formats: base, derived, and one I made up called traditional…see the mandi documentation for details), and \dirvect{} (think “direction of vector”) to get the symbol for a vector’s direction.
The arrow, and the vector it represents, has inherent properties that don’t change from one coordinate system to another, and some properties that do indeed change from one coordinate system to another. I want to do two things: 1) make a solid connection between what I’m saying and what students see in the textbook and 2) get them to think deeply about geometry. Where do the numbers associated with vectors in the textbook come from? They come from projecting a vector onto a coordinate basis. What does THIS mean? Operationally, it means the following:
- Establish an otherwise arbitrary orthogonal coordinate system, which I drew on the board apparently arbitrarily oriented around the arrow representing our velocity vector. I intentionally kept the arrow in the first quadrant though. The arrow’s tail need not be at the origin.
- Place an imaginary light source “far away” above the x-axis and let it shine down on the arrow with light paths that are parallel to the y-axis so that the arrow casts a shadow onto the x-axis. The length of this shadow, along with this shadow’s orientation, tell us “how much of the arrow lies in the x direction.”
- Place an imaginary light source “far away” above the y-axis and let it shine down on the arrow with light paths that are parallel to the x-axis so that the arrow casts a shadow onto the y-axis. The length of this shadow, along with this shadow’s orientation, tells us “how much of the arrow lies in the y direction.”
- Do a similar procedure to figure out “how much of the arrow lies in the z direction” but understand that it will be zero in this particular case because our arrow lies in the xy-plane.
- These three “how muches” are the numbers that go into the slots in the notation M&I uses to denote the coordinate representation of a vector.
Now, when I then said that each of these “how muches” has a technical name, and that this name is “the projection of the arrow onto an axis” there were audible “Oooos” and at least one “AHA!” because they said they’d heard this term before but didn’t truly understand what it meant until just now. I assume they were telling the truth, and I was happy. At this point, I did not distinguish between “component” and “projection” as the math textbooks do, and I will return to that later.
Next, we did a calculation to get algebraic expressions for the three “how muches” and noted a pattern: the “how much” along a particular coordinate axis always works out to be the arrow’s length scaled by the cosine of the angle between the arrow and the axis in question. I introduced the notation 
with obvious extension to the other two coordinate directions. Note the use of double bars for magnitude, consistent with the students’ calculus textbook. One question that was raised was why is cosine, rather than sine, used. We then discussed how this relates to the geometry of the problem, which is how much of the arrow lies along a chosen direction, so we need the side of the relevant right triangle that is adjacent to the angle we know, and thus we need cosine. Yes, you could use sine but then you’d have to refer to the complement of the obviously relevant angle and we want to be conceptually consistent throughout. So, we will always use the angle between the arrow and the chosen coordinate direction and we will always want the part of the arrow parallel to this direction and thus we will always use cosine. Students accepted this. They took the bait, because consistent use of cosine here leads to the next important idea: dot products.
Next, I explained that this process of getting “how much of a vector is parallel to a chosen direction” can be framed as a coordinate-free geometry issue. Given any two arbitrary vectors 
and 
(represented by arrows of course, but now I’m referring to the actual vectors rather than their representational arrows), and without introducing a coordinate system, we can answer the question of “how much of one is parallel to the other” in an elegant way. I introduced the new symbol 
as the symbol for “how much of is parallel to with no regard for orientation.” Note that’s one symbol! Specifically, it’s just a symbol for a real number, an element of 
. The actual number is merely the multiplicative product of the two vector magnitudes scaled by the cosine of the angle between them or just 
, which requires visualizing them as arrows for simplicity. Think of the symbol as having two slots, each of which takes a vector as input and the complete symbol represents the resulting real number. Okay here’s where I’m setting the stage for something huge later.
This leads to three obvious extensions: 1) filling both slots with the same vector and 2) filling one slot with a unit vector (which we call simply a direction or, now, a basic vector) and 3) reversing the order of the two slots. Well, the third is interesting because it doesn’t change anything! Yes, reversing the two slots changes the way the overall symbol looks, but we get the same real number out. BOOM! We have discovered an elementary example of a symmetric tensor! The first is interesting because is gives us a geometric way of calculating the magnitude of a vector…or does it? It’s not incredibly useful at this point, but we’ll make note of it and file it away nonetheless. The second is the important one for the moment, because by dropping a basis vector into one of the slots (doesn’t matter which one, remember) we get the same “how much” number we got earlier from other considerations! Now we have a solid way to quantity what we mean by projecting a vector onto a coordinate basis. It’s just a dot product!
To follow up this discussion, students did a whiteboard exercise. I asked them to draw an arrow represent a velocity vector with an arbitrary magnitude of five units and a direction to the right on their boards. They quickly picked up on the fact that the actual length of their arrow didn’t matter as long as they labeled as having five units. Next, I came around to each group’s whiteboard and drew a new orthogonal coordinate basis on it and asked each group to “project the velocity vector onto the coordinate basis that I drew.” They got it! I mean they totally got it! The only difficulty was, as I have come to expect, someone’s calculator being in radian mode and not in degree mode. Using VPython for computations will permanently fix this problem by eliminating it entirely. Anyway, the amazing moment came when I asked them to calculate the vector’s magnitude in terms of the components they just derived and they all got five units back! Some were amazed. Some expected this. I was happy either way. The takeaway? Projecting a vector onto a coordinate basis changes the numbers we use to represent the vector the way the textbook does, but the actual vector itself doesn’t change. That’s geometry! That’s physics! That’s cool!
That huge thing I mentioned above? I didn’t explicitly go into this in class this week, but here it is. We can also think of a vector as a function (a linear function, to be precise) that takes as an input another vector and outputs a real number by doing a dot product. Misner, Thorne, and Wheeler operationally describe is as a machine with an input slot that takes a vector and an output slot what spits out a real number. Yep…a vector framed as a function that takes another vector and outputs a real number, the dot product. This is conceptually very simple, but not at all anything close to what is seen in introductory physics texts. I’m trying to change that. Why? Because by stringing together such vectors-treated-as-functions in a certain prescribed way, we can build objects called tensors. Tensors appear in introductory physics, but we rarely point them out and exploit them. I’m trying to change that. At some point, I will revisit our class discussion from this week and present a vector as one of these functions that takes another vector and spits back a real number.
I welcome questions, feedback, and constructive criticism.