 Suppose we try to find the angle between two vectors. Using the law of cosines, we'll need to find the magnitude of the one vector, the magnitude of the other vector, the magnitude of the difference between the two vectors. And when we do that, it turns out we come across the expression of the form x1, y1, plus x2, y2, and so on. And this suggests the importance of the following. Let x and y be vectors where the x-eyes and the y-eyes are real numbers. The dot product is given by this formula, and we can call it the sum of the component-wise products. So for example, we might find the dot product of these two vectors with four components. So we'll pull in our definition of the dot product, and we compute. We'll multiply our first components together, multiply the second components together, multiply the third components together, and the fourth. And we'll add these products. The dot product allows us to express some other relationships very simply. So remember, we can find the magnitude of a vector by summing the squares of its components. But notice that the radicand is what we would get if we applied the dot product of a vector with itself. And consequently, we can say that if the components of a vector are real numbers, then the magnitude of the vector is the square root of the vector dotted with itself. We claim that the dot product has the following properties. First, we'll assume that we're dealing with vectors of the same size with real components. Then the dot product of u and v is the same as the dot product of v and u. We have commutativity. For any real number a, a u dotted with v is the same as a times the dot product of u and v. This is sometimes called associativity of scalar multiplication. And u dotted with the sum v plus w is u dot v plus u dot w. This is sometimes called the distributivity of the dot product. And since this is a theorem, you know that these things are true. Well, let's try and prove a few things so I can retain my mathematician card. Now, you might have seen proofs in previous courses. And here's the thing. The purpose of proof in mathematics is often misunderstood. We often present proof as a verification of an important result, but the reality is no one ever tries to prove something they don't already believe to be true. So why bother? And there's at least three good reasons for proving things. First, proof requires us to properly define what we're working with. If I claim that all glerms are squalish and you insist that I prove it, I'm going to have to explain what a glerm and what a squalish is. The other thing that proof does is it reviews things that we should know. You can't prove something unless you review what you know about the thing. And from a broader perspective, one of the most important things about proof is that proof reveals things we might not have known. It allows us to discover new things. So one of the things we claimed about the dot product is the dot product is commutative. u dot v is equal to v dot u. Let's try to prove it and let's make it a proof for vectors with just three components. So as we try to prove things, it's useful to keep in mind definitions are the whole of mathematics. All else is commentary. One of the most important things about proof is it requires us to review what our definitions are. And in this case, our starting point is we want to prove something for vectors with three components. Well, what does that mean that our vector has three components? Well, this means our vector can be written as, well, how about u1, u2, u3, and v1, v2, v3. Another useful idea in proof and in life and mathematics is that you can always write down one side of an equality. We want to prove that u dot v is equal to v dot u. We can write down one side, u dot v. And definitions are the whole of mathematics. All else is commentary. We have u dot v here. And our dot product has a specific definition. It's the sum of the component-wise products. And so we can write out the sum of the component-wise products. Now at this point, proof becomes something of an art form. And what's important here is that because we're trying to write this proof down, we should pay attention to what things look like. The typeset u's and v's look very similar and your eyes are going to water and glaze over if you try to distinguish them. So let's use a different letter. So maybe we'll use vectors x and y and our components x1, x2, x3, and y1, y2, y3. So we can still write down one side x dot y and we can still write down the dot product. Now we'll introduce another idea of proof strategy. It's easier to get where you're going if you meet someone from your destination. And in this particular case, we would like to get to y dot x because that's the claim. The dot product is the same no matter what order you take the vectors. Well that's my destination so I can write that down here at the end. Definitions are the whole of mathematics. All else is commentary. One of the reasons that definitions are so useful is they work both ways. If I have x dot y, I can write down the sum of the component wise products. But if I have the sum of the component wise products, I can write down the dot product. And what this means is that with definitions and nothing else, we can go backwards. So y dot x, our definition says that that is the sum of the component wise products and we can go back a step. And let's take a look at this. On this line, we have x1, y1. On the next line, we have y1, x1. Now since these are real components, we know that x1, y1, and y1, x1 are the same thing because when you multiply real numbers, it doesn't make a difference. Well if it's not written down, it didn't happen. So let's make a note of that. Since the x i's and the y i's are real numbers, we can switch the order of multiplication. Now one important thing, proofs have to be able to be read front to back, which means we have to be able to start at the starting point and end with our conclusion. So let's check it out. We have two vectors with three components. We find their dot product. Because the components are real numbers, we can switch the order of multiplication and rewrite the dot product. But this is just the dot product of y and x. So our proof is a good one. Well, let's try to prove for vectors with three components the distributive property. So it's useful to remember definitions are the whole of mathematics. All else is commentary. So this particular problem requires us to talk about vector addition and the dot product. So we'll pull in those definitions and remember we can always write down one side of inequality. So let our vectors u, v and w be vectors with three real components and u dot v plus w is, oh my, look at the time. Well, fortunately you can do this on your own and I'll leave you to do your own homework.