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.