 Welcome back to our lecture series Math 1060, churginometry for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misildine. In lecture 31, we will conclude our topic of vectors in this lecture series with section 9.4, and we're going to talk about the dot product. What is the dot product? Well, as we talked about vectors previously in this chapter, we've talked about addition of vectors. We can add them together. We can subtract them as well. We really can't multiply vectors. We can multiply a vector by a scalar, but a scalar is not a vector quantity. Can we multiply two vector quantities together? Well, it turns out that in linear algebra, that is sort of a nuanced topic, and there are actually lots of different types of products of vectors. In this concluding lecture for chapter nine, we're going to introduce the most commonly discussed product of vectors called the dot product. And they're not very clever names when it comes to these vector products here, because we often denote or name them by their, how we denote them, right? So you probably learned from previous algebra classes, you can do multiplication by like taking like, oh, I have two times four. I mean, you can also do three dot eight. You can do two times six. There's lots of different ways of doing multiplication. And it turns out all these different vector products, they use all these different notations as well. So one of them, we always use the dot notation and we call it the dot product to emphasize it. It's sort of a silly mnemonic device, but be aware it's present in vector algebra here. So the dot product, which will be denoted as u dot v, where u and v are vectors. So let's say that u equals a comma b, that's its algebraic form, and then v equals c comma d, it's its algebraic form. Then the dot product of u and v will equal a times c plus b times d. So notice what we did here. So a and c, this is the horizontal components of the two vectors. So you're gonna multiply together the horizontal components. That's where ac came from. And then what about bd? Bd is the product of their vertical components, b times d, you get bd. So the dot product is the sum of the products of the horizontal with the product of the vertical. Now be aware that the dot product of two vectors, it's itself a scalar quantity. The dot product of two vectors is not a vector, it's a scalar, it's just a number. There won't be any direction to a dot product. So u dot v gives you ac, which is a number, plus bd was the number. It's sum of two real numbers gives us a number. All right, so let's practice the calculation here. So if you have three, four, dot two, five, what you're gonna do is you're gonna multiply it together, their horizontal components, three times two, and then you add to it the product of their verticals four times five. So you get four times five like so. Three and two gives you six, four and five gives you 20, and so the sum gives you 26, and that's the dot product of the two vectors, okay? Another example, let's take negative one, two, dot three, negative five. So multiply together the vertical, excuse me, the horizontal, that gives you a negative three, then multiply together the vertical, that gives you a negative 10 that adds up to be negative 13. And so algebraically, that's how we compute the dot product. It's a very simple calculation in that regard. Is there any geometric intuition to what's going on there? We'll visit that in a slightly different video. At the moment, I just wanted to get used to the calculation itself. What if you wanna do a calculation like six i plus three j dot two i minus seven j? Well, this is just of course, unit vector representation here, but the same principle applies. You take the dot product, you put together the horizontal component, and so you're gonna end up with six times two. You'll multiply together the vertical components, which we see by the unit vector j. So you add to that three times negative seven, and so that's gonna give you six times two, which is 12, three times seven is 21, so it's a negative there, 12 minus 21 is equal to negative nine. That's the dot product here. But we've saw previously with this ij unit notation that things feel more out of break with this notation. If an intermediate algebra student were to walk in right now, and they would look at six i plus three j, and they dotted with two i minus seven j, right? What they would see here is they might not know that i and j are vectors, they might think they're variables, right? And they would think, oh, I'm just gonna foil this thing, right? That's what your intermediate algebra student would do, in which case they'd get something like six i dot two i, they would get six i dot seven j, like so. They would end up with a three j dot two i, and then finally, oh, I don't know if I can squeeze it in there, we'll get a three j dot a negative seven j, like so. For which case then, if you do usual properties, so you get six times two, which is 12, and you're gonna get this i dot i, they would call that i squared, mind you, but it's like, well, there's an i dot i. You're gonna get this negative 42, right? i dot j, like so, six times seven there. You're gonna end up with this three times two six j dot i, like so, and then finally you get this negative 21 j dot j. That's what that intermediate algebra student would do, because that's just following the usual rules of algebra. Then they'd probably recognize that i dot j is the same thing as j dot i, they'd add those together, but there's a simplification that's important to notice here. If i is equal to one zero, then what about i dot i? Well, that would equal one times one plus zero plus times zero, for which I would just give you a one. So i dot i is just a one, so i squared in this case just gives you a one, so you get 12. Similarly, if you think of j, of course, as zero one, then you get the same thing when you take j dot j, like so, you're gonna get zero dot zero plus one dot one, which is one. So in the end, you end up with this negative 21. So i dot i and j dot j just disappear, right? What about i dot j? i dot j, like so, well, you're gonna get one zero dot zero one. That equals one times zero plus zero times one, that's just equal to zero. So i dot j just disappears and this dot product is also commutative, so you see that j dot i is the same thing. So you end up with this zero and zero, and so these middle terms just disappear and you end up with the 12 minus 21 that we did before. So what I'm trying to tell you here is that the dot product does satisfy the usual foil rule from an algebraic setting. Now by all means, you don't need to foil these things out. It's much simpler to just know that the middle terms are gonna cancel out. It's just the product of the horizontals and the product of the verticals. But even still, I'm trying to convince you that this vector algebra satisfies the usual rules of algebra that we're used to, associative property, commutative property, distributive property, therefore the foil method all applies here. It's just, these aren't just variables, there's some geometric meaning to things like i and j. And we'll talk some more about this geometric interpretation of the dot product in our next video.