Introduction to Vectors
If you were talking to a friend about the drive you made this morning, you might tell them something along the lines of, “I drove from my place to hear in about an hour, going east for the entire time.” Your friend, who knows where you live and has a rough idea of the route to get to where you both are, understands implicitly that you don’t really mean that you went and drove in one direction only for the entire hour. After all, you first had to exit the property of wherever you were parked, then you probably had to navigate through the neighbourhood of where you live, before finally getting onto a highway or a direct route to your destination. It’s only at that point that you actually began going east. Before then, you may have driven in all directions, for varying amounts of time.
In physics, we call this route a trajectory. It’s how you move from one place and get to another. In order to describe the specific trajectory of your drive, we would have to do a lot of specifying, since your trajectory was complicated. But we’ll start off much more simply. Bearing in mind the impossibility of it all, imagine you just had a single block moving in a straight line, like so:
In this case, it’s quite a bit easier to describe how the route of this block will change over time. You don’t even need to be that great with the mathematics of it all for common sense to tell you that the block will move a distance d from wherever it is now by multiplying the speed at which it is moving and the time it travelled. If the block was moving at $5m/s$ and went for $3.5 s$, we would expect for it to move a distance of $d = (5m/s)(3.5s) = 16.5 m$.
This is a simple, but profound idea. We can figure out how far something has travelled by knowing its speed and multiplying it by the amount of time it has travelled for. Simple, right?
Unfortunately, that’s not quite the whole story. As you will see, a lot of learning mathematics is about understanding the special case, before moving on and getting more general. It’s the same story here. In the above example, it was easy to figure out where the block will be after a certain amount of time. Just multiply speed with time! But what if I asked you to look at this:
It isn’t so easy to figure out how far the cannonball will travel in a certain amount of time, even if you know how fast it is to start with. Common sense will once again tell you that, no matter how hard the cannon fires upwards, the ball will eventually fall back down to the ground. As such, the speed the cannonball is moving will change over time. Therefore, we will need a more sophisticated tool to analyze situations like these.
That tool is the vector.
What is a vector? Well, it’s a question that means something slightly different depending on if you’re a physicist, a mathematician, or a computer scientists. For our uses, the most practical way of thinking about a vector is as an arrow. The straight arrows you saw in the above pictures can be considered vectors.
The two main parts of a vector are its magnitude and direction. Direction means exactly what you think it means. In the above examples, the vectors I drew were pointing in the general right direction. So when we were talking about the block and I said it was moving $5m/s$, it would have been more accurate to say, “It’s moving $5m/s$ to the right.
The magnitude of a vector simply means the absolute value of the quantity we’re talking about (we always write magnitudes to be positive). In the first example, the speed of the block was $5m/s$, so that was its magnitude. However, these quantities don’t have to be fixed. Consider again the cannon scenario above. As soon as the cannonball leaves the chamber of the cannon, its speed is changing all the time (and, for that matter, so is its direction). As such, it’s a good idea to remember that while we will start with vectors that don’t change with time, there will come a point in which it is useful to use time.
But how is this helping us with the mathematics?
Fine, I told you about vectors, but how do we describe these situations with vectors? Isn’t it awfully vague to talk about something like a direction?
You’re absolutely right. To help, we need to use a tool that began with Descartes and is invaluable useful today. It’s a way to visualize what is happening in a system, so we don’t have to resort to only writing down equations. I’m talking about, of course, the Cartesian plane, where you’ve undoubtedly seen how to graph things like $y = 6x$ and the like. But now, you’ll find that it’s a fantastic tool for studying vectors.
Let’s go back to the example with the block, but this time, I’ll overlay a grid near the block so we can describe the situation more accurately:
From now on, you’re going to start thinking about space as a sort of grid system. Here, there are two directions: the $x$-direction, and the $y$-direction. Of course, you can go in a combination of these two directions, as the cannonball was doing before. But now that we have this grid system, we easily describe both the magnitude and the direction of any vector. Notwithstanding my bad drawing, the vector above has a magnitude of $5m/s$, and its direction is along the positive $x$-axis. In general, we measure the direction (or angle) of the vector by starting on the positive $x$-axis and rotating counterclockwise until we get to the direction that the vector is pointing.
This is all well and good, but I know that many of you are observant, and might ask, “If we have a vector going diagonally, can we describe it by talking about how much it moves in the $x$ direction and how much it moves in the $y$ direction?”
The answer is yes!
This highlights an important part of vectors: we can decompose them into their constituent parts. While it might be difficult to think about a certain motion in three dimensions, it’s much easier to think of the motion with a component in each of the $x$, $y$, and $z$ directions. Then, we can calculate things along each of those directions, before finally packaging everything up in the end to give a vector description of the motion.
(I want to take a second to mention that, of course, you can generalize the notion of what a vector is and what kinds of things you can do with them. This generalization usually leads you to the area of linear algebra, but we will focus for now with regular vectors that you can visualize as arrows, either in two or three dimensions.)
We will slowly build up to some interesting interesting results with these vectors, but I want to specify that I’m going to try and bring the logic and thinking behind a lot of these concepts from the perspective of someone who is looking to do physics. It can be all well and good to take a mathematician’s or computer scientist’s approach, but we’ll leave that to the side for now.
At this moment, you might be wondering something along the lines of, “Are these vectors really going to be that useful in physics?” The answer is a resounding yes. It may seem like vectors are only useful for describing the paths of bodies being launched, but the truth is that vectors are used all over the place. We use them to describe the equations of motion of bodies, the way electric and magnetic fields work, and basically anything that you can describe with the notion of forces. So basically all of physics, really.
For now, I just want you to take this initial point away: vectors specify magnitudes and directions, and we can add and subtract them component-wise in order to figure out how vectors interact with each other.
Can You Teach What You’ve Learnt?
Teachers are usually regarded as people who are experts at their subject. They may not be literal experts (such as those who teach scientific courses), but they have usually enough experience to make them experts. This expertise is usually forged through many years of teaching and working with students.
However, I’ve found that it has been extremely helpful for me to revisit content through the lens of a tutor. By switching roles from the person who is learning to the one who is explaining the concept, I’ve found that it helps reinforce the ideas in my own mind. When I can teach a concept to another student, it shows that I know what I’m talking about.
For me, the main reason I find this helpful is that teaching another student can be tricky. Often, the way you see something in your mind is not at all how another person sees it. Therefore, when teaching you have the challenge of not only transmitting an idea to the student, but of translating it as well.
This is particularly pertinent for the students I help in secondary school. Since that period of my life is several years old, the way in which I attack problems or situations demanding the skill set from that time has cemented in my mind. However, I need to sometimes put that aside when working with other students and focus on how they see the situation.
It’s a challenge that I think is worth thinking about. When learning a new concept, ask yourself: “Am I comfortable enough with this concept that I could actually explain it to someone else?” Trying to answer this question in the affirmative is a good way of assuring yourself that you’ve studied well.
It’s easy to have enough knowledge that you “pass” through a test without being noticed for your lack of knowledge. However, it’s much more difficult to do when you need to explain the entire concept to someone who has no experience. There, you can’t fall back on the chance that you won’t be asked a question on the one thing you aren’t sure about.
You can pass through a test without being discovered, but your goal should be to be capable of explaining a concept.
Derivations and Feeding
When I do nearly any assignment for homework, I’ll make a rough copy of my work before writing the copy that I will hand in. I do it for a practical reason: while it may take extra time to write my work twice, the truth is that I often take a lot of detours on my first try tackling a problem. I go down dead-ends, make little mistakes here and there that need to be corrected, and generally do a lot of messy work. Once I get the correct answer, I can tidy all my work up in order to make my final copy as concise as possible.
This work well for handing in assignments, but unfortunately this strategy is often adopted too much while teaching students new concepts. Instead of giving students time to think about how to prove or derive a statement, teachers often give the instructions to the student at face value. Additionally, there’s no “rough” work given, making students sometimes wonder why a certain strategy is being used. I’ve heard many times the phrase, “Don’t worry, you’ll understand what I’m doing in just a second.” It’s a nice thought, but I think it creates an environment where students don’t have to think about what they need to do themselves. Instead, they just need to follow along as the teacher feeds them the steps and the answers. This might be easier, but I definitely don’t think it’s as good of a strategy in the long run as getting students to struggle and go down those wrong paths.
Along the same line, this tendency to always show the “final” work sends the message to students that this is the only way they should have seen the problem, and having any other ideas wasn’t going to work. However, that couldn’t be further from the truth! We need to send the message that rough work is essential in science, because no one is likely to get the answer right on the first try. You need to choose a direction and make some progress before you can decide if you’re going to reach your goal.
If we want students to be more engaged in the act of learning and taking notes while looking at a derivation, we can’t spoon-feed them the method to use. Sure, it will give them the way to do it in the future, but it’s not a good way to get them to learn the concept. What we need to be doing is encouraging them to try ideas, see if they work, and refine them. That’s the basis of intuition. You get it because you’ve seen many problems like this and you have a feel for what’s going to happen.
When I work with younger students in subjects like mathematics or physics, it doesn’t take much to impress them with my ability to quickly see through a problem and calculate things that would take them minutes. Just like any other student at my level, we often skip the use of calculators because it’s easier to just focus on the work we are doing and do the arithmetic in our head. The most prominent example of this, however, is in algebra.
I’m definitely no expert on the subject, but from my experience with other students, the thing that seems to confound them over and over is algebraic manipulation. Consequently, it takes longer for them to learn any other concept after, because they’re always using the algebra that was already tricky for them.
For example, I’ve worked with some students on the process of factoring expressions. In secondary school, this is usually done with trinomials such as $x^2+5x+6$. The goal is to get from that form to the factored form of $(x + 2)(x + 3)$.
To do this, there is a whole set of instructions that students copy down onto their memory aids for the test. When I worked with these students, the procedure is what they followed.
As they were doing that though, I was solving it in my head. We then compared answers, and I’m fairly sure it was surprising to them how fast I had gotten to the answer, as if I was able to pull it out from the page itself. They, on the other hand, were carefully going through each step of the procedure before getting the correct answer.
Both ways worked, but it seemed as if I had a faster way. I showed them how I thought of the procedure in my head, and I could tell that they probably weren’t confident in their own ability to do that in their heads. To them, they were only ever going to work it out on paper.
What I’m trying to illustrate here is the massive difference only a few years of practice can make. I wasn’t separated from them by that many years, and what was a whole new problem for them to solve was something that I could typically do in about ten seconds or less. It isn’t because I’m brilliant at arithmetic that I can do this. Rather, it’s because of all the work I had done for years to get the hang of it. I can do it all in my head now, but I remember doing countless examples when I was younger, trying to get the hang of it.
In most areas of physics and mathematics that you enter, talent isn’t a prerequisite. Instead, being willing to commit to hard work for a long time is what’s needed. Do that, and you’ll be able to eventually appear “flashy” in front of those who can’t do something as effortlessly as you can.
But be aware: this is true for virtually any domain in life. Athletes don’t pull off amazing plays out of only sheer talent. They have an enormous backlog of hours that were dedicated to improving in their sport, and which culminate to help the player make the amazing play. You will find few examples of people simply having so much talent that they can do everything with little effort. More common is that you will see the hard work that was done by just pulling back the curtain a little on that person and looking at their history.