 Welcome back to our lecture series Math 1060, Trigonometry for Students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Missildine. Lecture 28 starts our new unit of chapter 9 about vectors, which honestly is a very, very fun subject, a fun application of trigonometry. In fact, vectors with the relationship to physics and other parts of physical science really show some of the power that trigonometry can have for physics, engineering, etc. And so, assuming many of us haven't seen the idea of a vector before, let's define it formally. In chapter 9, we'll actually look at vectors from two different types of representations. This first one we're going to refer to as the geometric representation, and this is how one would normally think about vectors in like a physics or scientific setting, which leads itself very much to trigonometry and to triangle diagrams, which we'll see some of those later on in this lecture. Later on in chapter 9, though, we will represent these vectors in a more algebraic sense, which abstracts the notion of vectors a little bit, but makes the calculation a whole lot easier, and one will argue it is definitely beneficial in the long run. So, in the physical or geometric sense, what is a vector? Well, a vector is going to be a mathematical quantity with both magnitude and direction, and I want you to think of vector from Despichl and me, right? You know, what does he tell? What does he tell Gru at the beginning of the movie? He commits crimes with both direction and magnitude. And so, think of a little vector character at that moment, right? And so, what do these terms mean, magnitude and direction? Well, let's start with the idea of magnitude. When you have a vector quantity, there's some type of strength, some type of dominance to it. And one of the best ways to think about vectors right now is to think about a force that's being applied. So, how hard can we push something? So, if we're trying to move a heavy object, do we push it a little bit? Do we push it a lot? Do we push a super, super big amount there? How hard are we going to push it? Do we apply three pounds of force, 10 pounds of force, 100 pounds of force, or maybe think of you're lifting a heavy object. Can you lift a loaf of bread? Can you lift a small dog? Can you lift a boulder? These all have different magnitudes, different strengths to the force that's being applied in those situations. But vector quantities are more than just this number. The magnitude will often represent with a numerical quantity. There's also a direction to a vector. And you'll notice that as I was talking, I was drawing little arrow quantities to represent vectors. And that's something we often do all the time when it comes to vectors. We think of them as little arrows, but there's some meaning to these arrows, right? So, the length of the arrow is to suggest its magnitude. The longer the arrow, the stronger the magnitude. A shorter arrow, weaker magnitude there. But also, there's a direction to it, okay? Imagine we have some very heavy object right here. I'm just gonna draw a rectangle that's sufficient to us, right? And it sits on the ground, right? And what we wanna do is we wanna move it. Well, if we wanna move it to the right, it would make sense to apply a force in the right word direction, causing the object to move, right? It would slide along the ground if we had a sufficient magnitude, right? But if we apply that same amount of force, but in the other direction, that'll actually cause the object to move to the left as opposed to the right, which could be a problem. Or if we exert the same amount of force pushing it downward, we're actually not gonna cause any motion whatsoever because the normal force exerted by the ground will be stronger than what we're pushing right there. And so the direction of the force makes a difference. And therefore, when it comes to vectors, we have to consider the magnitude and the direction. And like I said, we draw these with arrows. Another example to consider besides force that's related to it in physics is the idea of a velocity vector. Velocity is related to the idea of speed where speed represents how fast an object is moving, but a vector quantity has magnitude, which in this case would be the speed, but it also has direction. So velocity is a vector quantity because it has a magnitude, which we call speed, but it also has a direction. And so think of some examples of driving a car. One individual could drive a car 50 miles per hour due north, right? There is a direction and a magnitude to it. The length of the vector is its speed, the velocity vector, that is, and the directions you drive north. But one could also drive north at 25 miles per hour, right? The direction is gonna be the same, right? If two vectors have the same direction, we would draw them as arrows that are parallel to each other. Notice these two arrows are parallel. They're going the same direction, but this vector right here is actually half of the length of the first one because the first one's 50 miles per hour, the second one's 25 miles per hour. And so we would draw half the length there to represent the change of magnitude. For the velocity vector, a change of magnitude represents the speed of the vector there. But hey, we could also go 50 miles per hour and we could drive due south, in which case we would have to draw the vector, same length, right? So the length of this vector is identical to the length of this vector right here, but the directions are different, right? Look at the arrowhead. This arrowhead's pointing up for north. This one's drawn southward, a downward for south, right? And you'll notice that vectors that go in opposite directions will have the arrowheads on the opposite side. But we can go in any direction if we want to. If we wanna go northeast, we can drive that direction. If we wanna go southwest, these are options as well. An important thing to remember about vectors is that we'll draw them as arrows, but when we draw a vector diagram, well, for this lecture series, we're gonna focus just on two-dimensional vector situations. Of course, in real life, if you talk about forces and velocities and other type of vector quantities, this often has to be done in a three-dimensional sense, which is not inappropriate to discuss in a trigonometric setting, but for the conventions of this lecture series, we will reside only in two-dimensional settings. So as we talk about these physical vectors, be aware that the location that we draw the vector is irrelevant. So you see these three vectors on the screen right here, this one, this one, and this one. You'll notice that each of these vectors I've labeled as vector V, they have the exact same magnitude. Notice that the length of every one of these vectors is identical. Their magnitudes are the same. Also, their directions are the same. To see that, maybe I'll think of their direction in reference to the horizontal, right? If you think of some type of angle theta right here, again, what's the direction with reference to the horizontal, you get the exact same angle theta right there. And this is why trigonometry is gonna be very important in the study of vectors because the direction really is a trigonometric idea. It has to do with angles, which is what we taught study in trigonometry. So the magnitude and the direction does not change if we move the vector to some different place in space, like we see stuff like so. So the exact location of the vector doesn't matter. So coming back to these force diagrams you often see in physics, whether I draw my force vector external to the object or maybe internal to the object right here, it doesn't really matter. The location doesn't matter as long as the magnitude and direction is the same. Let me draw it so that the magnitudes look the same. So you could draw this same vector in one of many different ways. It doesn't matter the location in space is irrelevant. It's just the direction and magnitude that determines the vector. So that's a vector quantity. It's a mathematical quantity with both magnitude and direction. A vector quantity is in contrast to what's called a scalar quantity where a scalar quantity has a magnitude, but there's no direction to it. There's no direction for which you apply the magnitude. So for example, speed is an example of a scalar quantity. If I'm saying I'm driving 50 miles per hour that tells you how fast I'm going, but it doesn't give you the direction. So that's the difference between speed and velocity. Speed is a scalar quantity because it's just a magnitude, but velocity is a vector quantity because it has a magnitude, which is speed, but it also has direction. And so that gives you an example of the difference between vector quantities and scalar quantities. When we're describing vector quantities, it's quite common to use a boldface font to write them, to distinguish them from scalar quantities. So if I'm talking about vector variables or vector quantities, I might call it UVW. The idea is that vector starts with a V and U and W are letters adjacent to V in the Roman alphabet there, so UVW. And it's typically written with a boldface font. This is in contrast to scalar quantities, which we might call them XYZ, ABC, whatever. We'll still just use alphabetic letters to denote these things, but it won't be bold. And so this indicates we have a scalar. Now, admittedly, when you're writing with a pencil or a pen, it can be very difficult to write in boldface font. So when people are writing, they often will write vectors with some type of arrow notation above them. So I might say something like UVW. Again, this is because boldface is very impractical to write. It's something easy to do on a computer, but very difficult to write. So you'll see, you might see me writing little arrows above variable names. That's to suggest that it's a vector quantity to help us distinguish between scalar quantities. But as we alluded to earlier with speed and velocity, vectors and scalars are related to each other. Every vector does have a very important scalar attached to it, it's magnitude. And so the magnitude of a vector V will be denoted using this notation here. It looks like the absolute value notation. So absolute value of V. This represents the magnitude. This is the strength of the vector. And in particular, this will be a scalar quantity. Now, vectors are not just these arrows that live in space. The importance of vectors is that we can add them together. And I want you to think about that for a second. We can add together arrows. What? I mean, that'd be like saying, what's Pikachu plus Volkswagen, right? Clearly, clearly the idea of addition, combining things together, there should be some meaning to it, not just some arbitrary terms in there. But again, vectors are meant to be quantities, right? Vector quantities. We think of them as numbers, a different type of number, a higher dimensional number, right? Whereas a scalar, you can think of as just as a one dimensional number. These vectors are really thought of as two dimensional numbers. But again, you could also talk about three dimensional vectors. But again, we won't do that in this lecture series whatsoever. And so because vectors are quantity, it makes sense that one could ask about vector addition. But as a vector is not just a number, we can't just add numbers together, so to speak. How do you add together arrows? And so the rule for adding together vectors follows what we call the parallelogram rule. And it gets its name because of the following diagram you see right here. So to add two vectors together, let's say we're gonna add the vectors u and v together, you orient the vectors so that the tails of the arrows are touching each other. So u and v emanate from the exact same vertex, okay? And so then taking a copy of v, you're gonna add a copy of v to u right there. So you're gonna add a line that goes in the same direction, the same length as v. We're gonna do the same thing for u as well. Copy u and draw it over here, like so. And so when you do all four of these things, you're gonna u, you get a v, you get a v, you get a u. So when you put all of these together, you form a parallelogram. That parallelogram has a diagonal. And so you draw the diagonal to form the sum of the two vectors right here. So let me draw this from scratch here. Let's say we have some vector u and we have some other vectors say v like this. Like so, you're gonna copy your vector, something like this. You'd copy the vectors, make your parallelogram, connect the diagonals, connect the dots on the diagonals, connect in the dots as college mathematics, mind you, in which case you're gonna get u plus v, which would then be the sum of the vectors. It's the diagonal of this parallelogram, all right? Now personally, when it comes to addition of vectors, I like to, I mean, you can do this parallelogram or you form the parallelogram, but I think in practice, it's a little bit easier to think of it as the following where you're just gonna concatenate the two vectors. You start off with any of the vectors, u or v doesn't matter. Take v, take u like so. And so what you're gonna do is you're gonna put the start of the second vector at the end of the first vector, like so. So you add this vector, you add this vector, and then this would give you half of the parallelogram, half of the parallelogram as a triangle or you could think of a parallelogram as just a double triangle, if you copy that triangle here. And the diagonal of the parallelogram would then be the third side of this triangle, like so. And this perspective is gonna be very useful for us because again, this is gonna connect the trigonometry to the types of things we wanna study with this vector algebra here. So to add together two vectors, you're gonna take the one vector, concatenate it with the second vector, and then connect the start of the first with the end of the second. That vector would then be the sum of the two vectors. And the reason we adapt this rule is because this is how it works in physics. If you have two forces acting on a object simultaneously, two force vectors, then the net force would be the sum of those two vectors working together, just like we have with the parallelogram rule right here. It's also important to represent the negative of a vector. So given a vector, we define negative V to be the vector with the same length, same magnitude, but it's pointing in the opposite direction. So you can see that right here. So we have vector V, its additive inverse would then be called negative V. The vectors have the exact same length but they're pointing in opposite directions. And they have the consequence that if you take a vector plus its inverse, that'll give you the so-called zero vector. The zero vector is the vector with no magnitude nor direction. That is to say it's the vector which has no length whatsoever. Like if you apply no force to an object, then it doesn't move. There's no arrow there. You just represent the zero vector as a point. So let me show you an example of this vector addition in practice here. A very simple example. Imagine a boat is trying to cross a river and it's trying to go a boat's crossing a river that runs due north, okay? The boat is pointed due east and is moving through the water at 12 miles per hour. If the current of the river is a constant 5.1 miles per hour, let's find the actual course of the boat through the water. So you see what's going on right here. Our boat is traveling due east at 12 miles per hour but it's a river, it's flowing. There's a flow from the current of the river which it's flowing northward, right? So as the boat is going to the right, the river is pushing it upward. And so the actual course of the boat would then be the sum of these two vectors, right? These two velocity vectors. And so the actual boat would go in this diagonal course through the river like so. And so can we find this vector here that represents the actual course? Now, you'll notice that because we're going due east in the boat but the river's going due north, then these vectors are gonna be perpendicular to each other. It forms a right triangle right here. And so we can actually figure out this actual course here. The actual course we can describe as a vector, let's call it V, right? It's a vector, it has a magnitude and it has a direction. The magnitude we get from the Pythagorean relationship. Notice that this side squared plus this side squared equals this side squared. And so we see that the magnitude of V is gonna equal the square root of 12 squared plus 5.1 squared. Like so, 12 squared is 144. 5.1 squared is gonna give us 26.01. This still lives inside of the square root. We add those values together. We get the square root of 170.01. And that gives us approximately 13 miles per hour. Like so. So because of the current of the river, the boat actually is going faster than its engine or a rowing or I don't know how it's going, maybe it's a sail. It's going actually 13 miles per hour. But in what direction? Well, the direction, well again, the boat's going east, the river's going north. And so maybe we measure the angle here. What's the angle above the eastward direction and the northern direction? We ask about that. Again, this is a triangle. We can use some basic trigonometry to solve the missing piece right here. So what we're gonna do is we know this direction. That'll be the adjacent one. We know this side, that would be the opposite side. So tangent seems appropriate here. Notice that tangent of theta is gonna equal 5.1 over 12. Therefore, theta is gonna equal arc tangent of 5.1 over 12. For which we consult our calculator here and we're gonna end up with notice, of course, that if you wanted to play around with this fraction, you could, 5.1 over 12, the same thing as 51 over 120, which is the same thing as 1740, which is all over 40, which is the same thing as 0.425. Again, we don't have to get into all of that. We're just gonna put this in our calculator here and we get approximately 23 degrees, 23 degrees as the direction there. So our boat is gonna be traveling 13 miles per hour and it'll be 23 degrees north of the eastward direction. So adding together the vectors has a very natural consequence when we study velocity, when we study forces, when we study other types of directional quantities in physics and other scientific settings.