 In this second part of the discussion of vectors, we want to continue exploring our description of motion in all dimensions. And in particular, the key ideas that we're going to learn in this particular section of the course are as follows. Vectors, as we've seen in the past, can be added and subtracted from one another. But it's not the only arithmetic operation that's defined for vectors. It's possible to also multiply them, and we're going to see that in this lecture. And we're going to explore one kind of multiplication, something known as scalar multiplication, which takes as an input two vectors, but as an output it returns a scalar number. The scalar product, or dot product, represents the degree to which one vector projects onto another vector and takes that projection and multiplies it by the length of the remaining vector. This is a more compact way to handle this, as we will see as we explore the dot product. So let's consider the nature of the dot product. There are actually two kinds of vector multiplication, but we're only going to deal with one of them right now. And this, as I have said, is the scalar product or dot product. It goes by a couple of names. This product, this multiplication of two vectors, is so-called because the scalar product of two vectors, by definition, will return a pure number, a scalar, a directionless number. Two vectors go in, but a number comes out. And the notation for the dot product, which involves a dot, as the name implies, is shown here. If we take vector one, shown over here in blue, and we dot it, or scalar product it, with v2, shown here in red, the dot product of these two will, by construction, return a single number, capital V. Now, the dot here is extremely important. There's another kind of multiplication that involves a crossing x symbol, or the times symbol. It is a very different kind of multiplication. And so notation is extremely crucial when you are talking about multiplying vectors. You can't use either notation, the dot or times notation, like you can with numbers. For vectors, dot means scalar product, and the crossing x or times symbol means something else. So we're going to stick to this notation. You should accept absolutely no substitution for this when you're talking about the scalar product of two vectors. Now, before we proceed, it's important to know some very basic ground rules that involve the dot product of the unit vectors. The unit vectors define directions along specific coordinate axes. And we're going to stick with two dimensions, although this is easily extended to a third, or even more, spatial dimensions. The rules are as follows. The dot product of i hat with itself, i hat dot i hat, is one. Now, you can see already that the dot product clearly plays some role in our fundamental definition of the length of a vector. After all, a unit vector has length one. We observe here that the dot product of two unit vectors, i hat with itself, is exactly equal to one, which is the length of the unit vector, i hat, squared. The square of one is one. So we can already see that the dot product must somehow connect to length. And in fact, the dot product of any vector with itself will yield the length of that vector squared. So that already shows you the deep connection between the dot product and the length of a vector. Similarly, j hat dotted into itself yields one. Now, here's the key feature of the dot product as regards vectors. The vector i hat points only and entirely along the x direction. The vector j hat points only and entirely along the y direction. The dot product of i with j is zero. And that is because as you will see, the dot product tells you something about how much one vector, i hat, lies along the second vector, j hat. In this case, for vectors that make right angles with one another, absolutely none of the length of i hat lies along j hat. And vice versa, the statement works in both directions. None of j hat lies along i hat. And so it is true that any so-called orthogonal vectors, vectors that lie exactly at a 90 degree angle to one another and by construction i hat and j hat do, no set of orthogonal vectors has anything other than the null value zero for its dot product. So i hat dot j hat and j hat dot i hat are by definition zero. So anytime you see i hat dotted into itself, you can replace that with the number one. And anytime you see j hat dotted into itself, you can replace that with the number one. And anytime you see i hat dotted into j hat or vice versa, you can replace that with zero. And these rules will come in handy in a moment when we take a generic vector like v one and we take its dot product with another generic vector like v two. It is also useful to know that if you have a number a, like a equals five or a equals one hundred and seven, and you multiply that number by a vector to give it a length. So for instance, you want to define a length along the x axis. So you take a and you multiply it by i hat. If you then take the dot product with for instance i hat, you can move the number a around with no penalty and no change in the result. So for instance, I can start by multiplying a directly times i hat and I can take that whole thing and take the dot product with i hat or any other vector. This is equivalent to instead having taken i hat grouped a with the second vector and done the dot product because it's equivalent to this putting a out in front of the whole darn thing and just doing the dot product of those two vectors, which is one. So this is equal to a. So this is equal to a and the second thing is equal to a and the third thing is equal to a. They're all equal to a. It doesn't matter where you put the scalar. The scalar doesn't participate in the dot product of two vectors. So dot product is a very special multiplication that only acts on vectors. Scalers can be moved around anywhere in the process and they do not affect the result of the dot product of two vectors. Now, with those rules in mind, let us go ahead and do a dot product of our vectors. We can read off the actual numerical components of the two vectors. For instance, by looking at the grid of numbers and looking at the vector v1, we can see that it has a length of 1.5 centimeters in the x direction and a length of two centimeters in the y direction. Now, I'm not going to write those numbers down in every step of what I'm about to do. I'm going to keep the notation somewhat compact. So I will refer to the x component of v1 as vx1, the y component of v1 as vy1 and so forth. We can also similarly see the components of v2. It has a length of 1.5 along the x direction and a length of one along the y direction. So we could put numbers in here, but I'm going to hold off on putting numbers in until the very end as you always should when you're doing algebraic manipulation. So I have my two vectors, v1 and v2, and I'm going to take the dot product. I start by writing out the components of each of the vectors. So in square brackets on the left, this is my two-dimensional vector, v1. It's got an x component, vx1, in the i-hat direction. It's got a y component, vy1, in the j-hat direction. Here's v2, written out similarly. And here's the dot product between the two square brackets, again representing the scalar product or dot product of the vectors. Now the dot product can be distributed. So for instance, I can take vx1 i-hat and I can distribute the dot product multiplication of that component by component with v2. So I can write vx1 i-hat dot vx2 i-hat. And then I can add to that another term, vx1 i-hat dot vy2 j-hat. And then similarly, just like multiplying two polynomials by one another, I can take the second component of v1 and I can distribute it, again, adding all of these pieces together. And when I do this, I get this relatively unpleasant looking equation. It seems like I've taken a nasty situation and I've made it a whole lot worse by doing this, but wait for it, we're going to get some shuffling around of numbers and vectors in a moment that's going to grossly simplify what just happened. And in fact, once you get the hang of this, you should feel free to take shortcuts in going through this calculation quickly. The greatest shortcut of all waits for you near the end of this particular lecture video, but algebraically, if you're going to do the dot product robustly and completely this way, you can learn to throw out terms that have i's and j's mixed up in them because they're going to go away anyway, as you'll see in a moment. So here are all four terms of this distributed dot product multiplication. Here's vx1 i-hat dotted into vx2 i-hat, as I originally mentioned. Here's vx1 i-hat dotted into vy2 j-hat and so forth. Well, now I've got scalar numbers, vx1, vx2, multiplying unit vectors, i-hat, j-hat, etc., in a dot product. And remember the lesson I just gave you, scalar numbers can be shuffled around anywhere that's convenient to put them as long as you still keep them in the multiplication group and then gang the vectors together into their dot products. So for instance, I could move vx1 and vx2 out to the left of the dot product of i-hat and i-hat. And in fact, in the next line, that's what I do. Each term, term by term in the four terms, I pull the numbers out in front and I group in parentheses the vectors that are to be dotted together in a scalar product. So I've got i-hat dot i-hat, i-hat dot j-hat, j-hat dot i-hat, and j-hat dot j-hat. And you're beginning to see a pattern here now. And if you again, you remember the rules of the unit vector dot products, i-hat dot i-hat is 1, i-hat dot j-hat is 0. None of i-hat lies along j-hat, and the measure of that is given by the dot product and it's 0. j-hat dot i-hat is 0. j-hat dot j-hat is 1. And so in the next step, I've got the first term multiplied by 1. The middle terms cancel out because they go to 0. They go away. And then finally I have the fourth term which just is multiplied by 1. So simplifying all this and taking all the clutter away, what I've learned is that the dot product of v1 and v2 is just the product of their x-components added to the product of their y-components. See the shortcut? The shortcut emerges at the end of this labor. The dot product of any two vectors is the product of the x-components added to the product of the y-components added to the product of the z-components. That's it. That's the rhythm of this. Now I can plug in numbers. The x-components of the two velocity vectors v1 and v2 are 1.5 centimeters and 1.5 centimeters. These are now just numbers multiplied together so this dot here just represents plain old scalar multiplication. And then similarly the y-components are multiplied together. The y-component of v1 is 2 centimeters. The y-component of v2 is 1 centimeter. And you see that written out here. And then I can finally do all the math and group all the stuff together. And I get my final answer that the dot product of v1 and v2 is 4.25 centimeters squared. Now notice what would happen. Just think about this for a moment. Maybe you want to work this out on paper. Not a bad idea. Instead of doing v1 and another vector v2, do v1 dot it into itself. v1 dot v1. Well what happens? We wind up here with vx1 times vx1 plus vy1 times vy1. So we have vx1 squared plus vy1 squared. Doesn't that sound familiar? That's the Pythagorean theorem. That's the sum of the squares of the sides of the triangle that make up this vector. And that equals the square of the length of the vector, the hypotenuse of the right triangle. So the dot product is just an elaborate way of recovering the Pythagorean theorem but from vectors, not just plain old numbers. But really, again, what the heck is this number represented by the dot product? So it's great that we got a number out of two vectors. Numbers are always nicer to deal with than vectors. But what is this number? In a sense, it represents the length of one vector that projects onto another. What it really is, as you'll see in a graphic in a moment, is it's the component of one vector, say vector a, that lies along vector b multiplied by the length of vector b. So if we think about the unit vectors and the rules, again, about multiplying i hat and j hat with itself or with each other, we can begin to build some physical intuition about what the dot product is. Again, the expression i hat dot i hat kind of asks the question, well, how much does a unit vector pointing in the x direction project onto itself? Well, of course, the answer is all of the unit vector pointing in the same direction as another unit vector lies on the other unit vector. The answer is 100%. And sort of that's what this quantity returns to us as a number. A vector projected onto itself always equals the full length of the projection. In this case, two unit vectors, really, this is the length squared in this case. The same goes for j hat dot j hat. And then as I said earlier, the expression of i hat dotted into j hat asks and ultimately answers the question, how much does a unit vector along x project onto a unit vector that lies instead along y? And the answer is not at all. None of a unit vector pointing on x has anything to do with y and vice versa. So lines or vectors at right angles to each other have no degree of their length projected along the other. Remember, ideal geometric lines have length in only one dimension. They have no dimensionality in any others. When we write a line on a piece of paper, we're approximating an idealized geometric line by giving it a thickness. But in reality, a geometric line has no thickness. It has length along one direction and no dimension in any of the other two dimensions of space. But that's sort of throwing yourself back to geometry a little bit to remind yourself that a vector is a line and a line of dimensionality in one direction. There is no dimensionality in any other direction. Now, a shortcut for the dot product is this rule of thumb that in general, the dot product between any two vectors, a vector and b vector, whatever they may be, is given by the product of their lengths, a and b, and the cosine of the angle between them, theta. So if you instead know the angle between two vectors, don't spend all your time on that elaborate dot product I just showed you. Sometimes you'll be given vectors where you have no choice and you have to do that. But if instead you either can figure out or are given the cosine of the angle that is directly between those two vectors, just take the length of the first times the length of the second times the cosine of the angle, quit and go home. You're done. So the key idea is that we have learned in this section of the course, this brief discussion of one kind of multiplication for vectors is as follows. Now, vectors can be multiplied. They don't just have to be added or subtracted. There's one kind of multiplication that we've just looked at. There's another kind we'll get to eventually, but not right away. And that this kind of multiplication, scalar multiplication, takes as input two vectors and it returns a scalar number. The scalar product or the dot product, which implies the symbol you should use when you write it down, represents the product of the projection of one vector onto another vector and the length of that vector. So that's represented over here. We can see graphically the dot product represented for us. We have the vector A. We have the vector B. We have the angle between them, theta. The dot product is AB cosine theta, where B is the length of the vector B. A cosine theta is this. It's this length right here. And that's part of the components of the vector A composed along the direction of B. So if I take B as if it was the x-axis and I write my horizontal component of A and my vertical component of A, the horizontal component is given by A cosine theta. This is something you've been exercising over and over and over again in this course as you look at velocities, accelerations, forces, et cetera, and do some trigonometry to relate forces to each other and add them up and so forth. Here you can see that the dot product is the length of the piece of A that lies along B times the length of B. So it's A cosine theta times all of B. Similarly, you could instead decompose B along the direction represented by A and there also the corresponding component that lies along A is B cosine theta. That doesn't change. And then you multiply B cosine theta to answer. So the dot product works whether you start from A and go to B or start from B and go to A. And that's one of the beautiful things about this product. And at the end of the day, two vectors go in, one number comes out and now you have a number that characterizes the sort of degree of overlap of these two quantities. It also as you will see in a physics context has specific physics meanings. We're about to encounter the dot product in one special location in energy and specifically in work. But it's not the only place that it appears. You will see it in the study of other things like electromagnetism. And so it's a good idea to begin to build some familiarity with the dot product. And of course you're going to get to do that in this course as we apply this concept and physics concepts to understanding the natural world.