 Welcome back to our lecture series Math 1060, trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lecture 30, we're going to continue to talk about vectors. In section 9.3, we're going to talk about the algebraic representation of vectors. Now, I don't usually draw much attention to the really obscuring goofy titles to our lectures here, but I figure since it's named after me, I should probably say something about this. You might have remembered in lecture 28, when we started this topic of vector discussion, it was referred to as Robin Hood's favorite section. Well, that one makes sense. I mean, most of the jokes in this lecture series are completely obscure and no one gets them except for me. But that one makes sense. Vectors are arrows, Robin Hood is a famous archer and such. But what does this one have to do with anything? Well, it turns out that vectors, the topics of vectors is really the introduction to a broader discipline of mathematics referred to as linear algebra. With regard to the lectures I have posted here, my linear algebra lecture series is in fact the flagship of them all as it's based upon my own textbook about linear algebra. It's called Linear Algebra Done Openly. Let's just say I have a great passion for vectors, especially the algebraic representation of vectors. I think you're going to see that in this lecture and the subsequent videos for this lecture as well. What is so cool about the algebraic representation? Well, it turns out that while vectors are naturally defined in a trigonometric setting, as we've learned previously in this lecture series, that there's a difference between algebra and trigonometry. Trigonometry does some things better than algebra, but algebra does some things better than trigonometry. It turns out when it comes to vectors, even though it makes sense to define them trigonometrically, when it comes to actually doing algebra with them, it makes sense to do this algebraically. We're going to start to see this in this lecture. What is the algebraic representation of a vector? Well, let V be some vector, right? Let's put this vector in standard position. Remember, the location of a vector in the plane doesn't make any difference on the vector's value itself. It doesn't change the vector quantity. Standard positions when we put the tail of the vector, think of it as an arrow, at the origin, so at the point zero, zero. Well, and you can see that illustrated in this diagram right here. Well, if the vector is put into standard position, it's tail is on the origin, where is the head of the vector pointing to? Well, there's going to be some unique point in the plane when you're in standard position. Let's call that point a comma b, a is the x-coordinate, b is the y-coordinate. In which case, then, this point's uniquely determined by the vector. If we took a different vector, it would be pointing to a different point. Different vectors give us different points in the plane, right? And so we could characterize the vector by using this point, and this leads to the algebraic form of the vector. So we'll say v is equal to bracket, angle bracket a comma b, angle bracket here. We're using these little brackets here to distinguish it between the point, because we're not talking about the point in space. We're talking about the vector, so it's the arrow that's pointing to that point. Now, these numbers a and b are not just some arbitrary points, not arbitrary numbers in the plane. This is actually, these are the components of the vector v, and so the algebraic form is sometimes called the component form. That is to say the horizontal component of v is just this number a, and the vertical component of v is just this number b. And we can see this from the usual Pythagorean equation, because we, after all, the horizontal and vertical, they form a right angle with each other. They're perpendicular. And so think about the coordinate a, right? This is the shadow cast on the x-axis. That's what this number a is, and b is the other way around, too. It's this vertical component. How far up the y-axis do you go? That gives you b. So these two numbers, again, are not just arbitrary numbers that v just so happens to be pointed to. These are the components of the vector. And therefore, the translation we had previously, the conversion formulas we have are applicable when we go from the geometric to the algebraic representation. So for example, because these three numbers, the magnitude of, I mean, a is the magnitude of the horizontal components. So this number a, this number b, and the magnitude of v, they form a right triangle. So these three numbers are related to each other by the Pythagorean equation. a squared plus b squared equals the magnitude of v squared. Or if you solve for the magnitude of v, you get this is equal to the square root of a squared plus b squared. Similarly, if we think of theta as the angle between the positive x-axis and the vector, which is an arrow itself, this theta represents the direction of the vector. Well then, if you take opposite over adjacent, there's a tangent relationship between the angle and these components, tangent theta equals b over a, or you could take arc tangent and bosa as well. And it goes the other way, it goes the other way, of course too, that a is just equal to the magnitude of v times cosine of theta, and b is equal to the magnitude of v times sine of theta. So we can convert back and forth between the geometric and algebraic forms of a vector using these trigonometric formulas, all right? So let's do a quick example of this. If we take the vector three comma negative four, what does that mean? It means that we're gonna draw the arrow that points to three negative four. x-corner is three, y-corner is negative four. You see a vector in the fourth quadrant right there. How do we find its magnitude? Well, that's easy enough. The magnitude v is gonna equal the square root of three squared plus negative four squared. Three squared is nine, negative four squared is 16. Nine plus 16 is 25, and we have our favorite Pythagorean triple. We see that the magnitude of our vector v here would be five. And so conversion from the algebraic form to the geometric or the geometric to algebraic form of a vector is just these basic Sokatoa right triangle trigonometric relationships, all right? So why do we care about the algebraic form? I said earlier, it makes things easier. Get to that point. All right, I'll get to that. So previously we talked about the idea of vector addition. We can add arrows together, and this makes sense that these arrows represent forces or displacement or velocity and things like that. The way you add together vectors geometrically is by the parallelogram rule, which you see a picture of such a thing down here. We have some vector u, we have some vector v, and so you can add them together so that the sum u plus v is the diagonal of this parallelogram diagram. Okay, that's how you do it geometrically, but how do you do it algebraically? It turns out it's so, so, so much easier to do this algebraically. If you wanna add together u plus v, take their algebraic forms. So u has some algebraic form, so it has a horizontal and a vertical component called that u x, u y, very clever mnemonic device there. And then v also has an algebraic form. Call its horizontal component u x, and it's, excuse me, v x, and call its vertical component v y. How do you add together the vectors? Well, you're just gonna add together their horizontal components. You're gonna take u x plus v x. That'll be the horizontal component there. And then how do you add together its vertical component? We'll take u y plus v y, and that's its vertical component. So use add together basically like terms. You add together the corresponding components, and that's how you add the vectors together. So let's see an example of this. Let's say u is equal to six two, and v is equal to negative three five. How do you add them together? Well, it's easy enough. You'll add together the horizontal components, six minus three, add together the vertical components, two plus five, and you end up with six minus three is three, and two plus five is seven. And so the sum of the two vectors will be three seven, which if you draw these vectors in standard position like you see in this diagram, u is six two. So it's gonna be the vector that goes to, it's gonna point to the point six comma two, x is six, and y is two. Now we can draw v in standard position as well, right? This is gonna be the vector that points to the point negative three comma five like you see right here. But to make life a little bit easier, I'm gonna translate it over here, right? And so this is gonna be the vector that goes to the left by three and goes up by five. And so think about that for a second. We go to the left by three, one, two, three, and we go up by five, one, two, three, four, five. We end up at the point three comma seven, and that justifies why we add these things together. And so algebraically, we just add or subtract the corresponding components, and that gives us the vector sum. That's a lot easier than this parallelogram rule for sure. Let's introduce another operation for vectors. It's very common that we've alluded to in the past, but never done it explicitly. The idea of scalar multiplication, we can multiply a vector by a scalar, and multiplying a vector by a scalar, what it does is it just multiplies each of the components of the vector by that scalar. So for example, if you take v equals three two, then what's three v? Three v just means that you're gonna times three by the components of the vector. So you're gonna end up with three times three and three times two. So you end up with nine comma six, like so. And so in this diagram, you see the two vectors in play here. They're both in standard position. There's vector v right here. v was three two, right? So it goes three to the right and then two up like so. Three v by the formula, you're gonna get nine six. So you're gonna go one, two, three, four, five, six, seven, eight, nine to the right. And you're gonna go one, two, three, four, five, six upward like so. And so three v is this vector right here. Now when you look at three v, it goes in the exact same direction as v. The direction didn't change. This angle is still theta, whether you're talking about v or three v, but the magnitude's definitely bigger. In fact, the arrow looks three times as long as we started off with. And this is what scalar multiplication does. Scalar multiplication will stretch or compress the length of the vector, depending if the scalar is large or small. So times in v by three may to get three times longer. If I times v by one half, it would cut the magnitude in one half, keeping the same direction. If your scalar is negative, then scalar multiplication will flip it to the other direction. So like v times negative one would give you something like this. It flips the direction. But scalar multiplication is easy to do when it comes to algebraic forms, because you just times each of the component by that quantity, but it has a geometric consequence. Scalar multiplication stretches or compresses the vector, but it's easy to do algebraically, okay? Some other things I should mention that the zero vector, this is just a point in space, right? The zero vector has the algebraic form zero, zero. Notice if you add the zero vector to any other vector, it won't change that vector. Because the zero vector is in fact the additive identity of vector addition. Well, what about the additive inverses of vector addition? How do you do vector subtraction? Well, if you have v equals a comma b, the negative v is just gonna equal negative a times negative b, because this is just negative one times v. Like we said on the previous slide, times b by negative flips it in the opposite direction. And so algebraically, we can very easily capture this idea of additive inverses. All right? Well, let's do some slightly more complicated calculations. Take u this time to be five and negative three, and take v to be negative six and four. How would you compute u plus v? Well, I don't have to worry about a parallelogram diagram or anything like that. If I wanna do u plus v, all I have to do is just add together the corresponding components. Horizontal with horizontal, vertical with vertical. Like so. And so we get five minus six, which is negative one, and we get negative three plus four, which is positive one. So the sum of the two vectors is negative one one. No diagram necessary, no geometry if it's in algebraic form. Well, what about this one right here for u minus five v? What if we throw some scalars into this sum as well? This object right here is commonly referred to as a linear combination. A linear combination meaning that we take some scalar multiple of the one vector plus some scalar multiple of the other vector, which admittedly u plus v is a linear combination because your scalars are just one in that situation. But this is sort of like the most general way of combining two vectors together. You can scale them and you can add them or in this case, attract them. But the calculation is no more difficult than what we've been doing previously in this algebraic form. You're gonna get four times u, which remember u is five and negative three. You're gonna take negative five times v, which is negative six and four. So do the scalar multiplication. So for the first one, you get four times five, which is 20, and you get four times negative three, which is negative 12. I want you to think of it as like when you have this vector you can distribute the scalar onto each of the pieces. And then for the next one, you distribute the negative five. You end up with negative five times negative six, which is positive 30, and negative five times positive four, which is negative 20. And now you add these things together. You're gonna get 20 plus 30, which is equal to 50. And then you're gonna get negative 12 plus negative 20, which is negative 32. And this would then be the linear combination, the combination for u minus five v. So vector algebra is super easy when you're in algebraic form. Adding and scaling vectors is a cinch. As long as we can convert from the trigonometric form to algebraic and back again, we can use the simpler algebraic form to help us solve problems related to vectors. And this idea of translation is a big deal, right? Trinometry is sort of the native language of vectors, but the more efficient language is algebra. So if we can convert the vectors, we can make life so much easier with these vector applications.