 Vectors. Now look gang. I'm gonna ask you a question. How fast can you drive on the highways where you are? All right? How fast can you drive? Green's theorem. As in the 3D, yes, should have clarified but probably better to stick to yeah. Yeah. 65 is the max in US. Really? I think in the US there's places where you can go 80 as well, no? Just for funsies. 100 kilometers per hour. Yeah. Right? Or 110 kilometers per hour. Here in Germany, unlimited indeed. The autobahn, Lerking, Loris. 120 kilometers on the on the biggest ones. Cool? I can drive pretty fast. 70 miles per hour. Divide 70 miles per hour. No. We have 70 in Pennsylvania. Yeah. I'm pretty sure there's places you can go 80 in this US if I remember correctly. So that is a distance. That is a scalar quantity. How fast you can drive. Right? Now if you put a direction to that, that becomes a vector. Okay? So if you say I can travel 70 kilometers. Let's make it what do you guys want to do? Kilometers? The whole world does kilometers. The US does miles but the majority of my audience in general is American. So things 120 in Ireland. By the way, can you do this kind of math? Because it looks crazy. Yeah, for sure. That's just a distribution. Should we do that one? Speedy Gonzalez style? Let's do vectors first and then we do that one. Right? So take a look at this thing. Let's do simple 100 kilometers per hour. Let's say you can go 100 kilometers per hour. Right? That's a scalar quantity. Okay. Montana is 80 now. Montana is 80. Cool. Yeah. Montana. I've driven through Montana. Nice try. Used to be reasonable and prudent. So this is scalar. Scalar. Right? Which means there is no direction to it. But if I go 100 kilometers per hour north. This is a vector quantity. This is a vector because it has direction. Okay. You're a good guy, man. Thanks. I tried to be. Right? So that's as soon as you add direction to force, to motion, to anything, you end up with a vector. Right? So there are two main types of vectors we look at. One is motion. The other one is forces. Right? And when you're adding vectors together. So for example, let's assume we had this. We had, let's say we had 100 plus 30 plus 80 minus 60. Actually, that's not the at the minus. Let's keep it simple. This. If you're adding these together, then this is 120. Sorry, 120. 210. Right? That's 110 plus 100 is 210. So that's 210. Right? If this is apples, then it's 210 apples. If it's kilometers, then it's 210 kilometers. Right? Make sense? Okay. Now, if these things had direction associated with them, it becomes a vector. You can't just add it up that way. For example, let's assume this was 100 North plus 30 Northwest plus 80 East, Southeast, Southeast. Southeast. Let's say we wanted to do this. Right? You can't just add the numbers because the way this looks is this. Take a look. Oh, algebra in college. I needed a waiver for it. My eyes just went across. So take a look at this thing. Let's say we have 100 North. So 100 North would be here. Right? Let's assume that's 100 North. And then we have 30 Northwest. Here, let me draw this better. We'll tag on to it. Right? 30 Northwest. So you want 100 North, 100 North. And then when 30 Northwest, let's assume that's 45 degrees. Right? So we want 30 Northwest. Let's make the vectors proportional. If that's 100, right? Then a third of that would be 30. So 30 Northwest and then Southeast. Let's go Southwest. We can draw it properly. Southwest. Okay. There's South and then West. And then you went this way. 80 this way. 80 Southwest. Right? This is where you end up. You start off here. This is where you end up. If you're going to add all those together, the total of those is going to be from there to there. That's what it adds up to. Right? So this plus this equals this. So because this is a vector, there is a magnitude involved here. Magnitude, magnitude plus direction. So it's got magnitude plus a direction. How do we figure this out? Looks complicated. If you have a ruler and a protractor, you could do it accurately. Make centimeters represent, well, that would be a meter. Make millimeters be one unit. Right? And then draw it and have protractors going on and stuff like that. Well, to do this algebraically, we need to break this down into its components. And when it comes to components, we're talking about the Cartesian coordinate system. X and Y axes. X and Y axes. Why is that the case? Because if you're adding a vector that's going exactly in the same direction as another vector, then you can add them straight up as if they were scalars because they're acting in the same plane. Right? So if you had this, you had 100 North plus 30 North plus 80 North, then that would be 210 North. It's the same unit. Right? It's just like 100 apples plus 30 apples plus 80 apples equals 210 apples. 100 North, 30 North plus 80 North is 210 North. Right? But if they're all going in different directions, then what we need to do, we need to break them up into components. Right? Let's do one. Let's do one. That's the intro to the vectors. Right? And the best way to appreciate how this works, we need to do an example. And you need trigonometry, so couture to be able to do this. Okay. Now watch. We're going to draw this. Okay. Now we're going to draw this. I need to, I need, let's see what kind of pens we got here. Let's see what we've got. I got a lot of pens out here right now. Hopefully I'm not going to drop them to make nasty noises. Oh, look at this. Let's make a Cartesian coordinate system. Okay. Let's make it, no, let's make it up here. How's that come out? Not bad. I wish they made dry erase pens that lasted longer. These things die off so fast. So once we've got a Cartesian coordinate system, let's add what we had. Take a look. We had something that was going 100. So these are the measurements we had. We had, where should we put this? We'll just put it here. 100 North. We had 30 North West. And then we had 80 Southwest. Southwest. Right? So let's do the 100 North first. So 100 North, lamb ex, exitex, excitex, lamb excitex. Hello. What if you change the place of vectors in the sum? What if you change the place of vectors in the sum? It wouldn't matter. It wouldn't make a difference. You'll see. Take a look at this. So we're going to break this thing down into components. So let's put them all originating from the zero. So we're going to go 100 North is here. So let's assume this is 100 North. So that's going to be 100. The next one is 30 Southwest. Oh, sorry. Northwest. Right? 30 Northwest is going from here. That's Northwest like this. Let's make the 100 a little bit bigger. That way we can do the work 100. And this is 30 this way. Okay. And then the other one is 80 Southwest. Southwest. And 80 would be about this far, this far. So we're going to come here. So this is going to be 80. Now, if I say Northwest, I'm going to assume it's 45 degrees. So let's assume this is Northwest means 45 degrees. Okay. That means this angle here is 45 degrees. Five degrees. Okay. MC Mike. Okay. Makes sense now since the triples relate to volume. Yeah, yeah, yeah. For sure. For sure. And because I'm pretty sure you're doing integration, you're trying to figure out the volumes of things, right? Maybe MC Mike. And again, Southwest will assume this is 45 degrees. Now remember, this could be any degree. If you want here, let's make it, let's make this here. We'll redraw this. Let's assume it's this. Actually, let's do it this way. That way. And usually, usually you want to do your measurement to the x-axis, the angle. So let's assume this would be Southwest. If I say Southwest 60, then this is 30. Okay. Because this part would be 60. South 60 degrees West. There's different ways of saying this, right? I'm just writing it like this. I'm explaining to you. There's different ways to represent this. I think they go South 60 West. But I don't want to use the structure of saying things without it being accurate, right? I'm not going to assume this is the way they write it everywhere. Okay. So right now, because that's 60, I'm going to keep that as 30. Okay. Because I want to just do everything relative to the x-axis. Okay. And this was 100. Remember, this was 100. This was 100. This was 100. And this was 80. So what you want to do is put them in the same direction to be able to add them, right? Because again, if this was 80 going down, then the total sum would be 100 minus 80 because it's going the opposite direction, right? So what we end up doing is we take each one of these vectors and break them into x, y components. So let's take the 30. Let's do it here. Let's take this guy. 30. And we're going to break it down into components in the x direction and the y direction. Okay. So if that's 45 degrees, 45 degrees. Okay. So what we want to do now is figure out what this is and figure out what this is. But we're also going to do this with this. So I'm going to call this, what should we call this? x1 and y1. Actually, let's call them x2 and y2. x2 and y2 because the 100 is really our first vector. Okay. So if we're going to do this, we need trigonometry. Right? So if we're going to use trigonometry, we're going to use, let's see how dark this one is. So we're going to go, if you remember Sokotoa, Sokatoa. Sine of an angle is opposite over hypotenuse. Cos of an angle is jason over hypotenuse. Tan of an angle is opposite over jason. And we usually just end up, actually it comes out not too bright. I'm going to use black still. You're basically going to use sine and cosine because you're looking for x and y. So check this out. If you're going to use that, let's try to figure out what x is. If you're going to look for this, that's adjacent to 45 degrees. So you're going to use this. So you're going to go cos of 45 is equal to adjacent, which is x2 over 30. So x2 is going to be equal to cross multiply up 30 cos of 45. If you want to find out the y component, you're going to go sine of 45 is equal to y2 over 30 cross multiply. So y2 is equal to 30 cos of 45. Let me highlight this because these are the ones we're going to use. So we just broke this down into its component here and here. That's what their links are. Let's do the same for the 80. So 80 goes down and here is the x component goes in this direction and the y component goes in this direction. So this is going to be 80 and this is 30 degrees. And this is x3 and y3. Now if you want to keep track of your variables, you could call this x80 and y80 referring to the 80 magnitude vector and you could call this x30 and y30. I'm just calling them two and three just because it's simple right now. We don't have too many variables I want to keep track of. Well, if you use Sokotoa again, then you have cos of 30 is equal to x3 adjacent over hypotenuse. So x3 is equal to 80 cos of 30 and sin of 30, if you're going to use the y, is going to be y3 over 80. So y3 is going to be equal to, what are we, 80 sin of 30. I hope that's clear. I'm sort of going through this. I'm assuming you know trigonometry in this. I'm assuming you know trigonometry. Now take a look at this thing. Okay, so we have the x part of this guy. We have the x part of this guy too now, right there. We've got the y part of this guy and we've got the y part of this guy now and we have the x part of the 100 because there is no x part. It's just going straight up. So the y part of 100 is 100. So if we're going to add up these guys, if we're going to go this plus this plus this, then we have to add the components. So the x part, x part is going to be x total, let's call it x total. Let's see if we're going to, should I do this in red? Let's do this in red. Red is coming out okay. x total, x total is going to be the x part of number one. So x part of number one plus the x part of number two, which is right here, x two plus the x part of the 80, right? Well the x part of number one is zero plus because it's not moving in the x direction. It's only moving straight up in the y direction. x part of number two is this guy, 30 cos 45 plus the x part of the third part which is 80 plus 80 cos 30. And what you can do is you can punch that into your calculator. Now if you know your special triangles, 30, not 30, actually we've got to do 30 anyway. So 45, 90, 45, 1, 1, root 2. And then the 30 triangle is 30, 60, 90, 1, squared of 3 and 2. So cos 45, cos 45 is 1 over root 2. So this is going to be 30 times 1 over root 2, oops, squared of 2, plus 80 times, what's cos of 30 degrees is root 3 over 2. Root 3 over 2. So this is going to be equal to 30 over root 2 plus 2 goes into 80 40 times 40 squared of 3. That's the x part of the total component. Now you wouldn't need to do it this way, you would just punch into your calculator, right? So let me move my pens here. So let's punch this thing because we need the numbers. Let's do this. What have we got? I'm just going to punch decent. 30 cos 45, 30 times 45 trig cos equals, so this part is 21, 21, 21.21 plus 80 cos 30. 80 times 30 trig cos equals 69.28, 69.28, which equals, and you add these guys up, plus 21.21, 21.21 is equal to 90.49, 90.49. That's the x direction, okay, which makes sense. Take a look at this thing. This part added to this part will make it go this far, right? That's what really we're doing. Okay, does that make sense? 21.21 is this length here, 69.28 is this length here, and this one contributes zero in the y direction, in the x direction. So if you add this and this, you get this, 90.49. Now let's do the y part. If we do the y part, I'm going to erase this part. Let's do the y part. y total is equal to y1 total plus y2 total plus y3 total. Well, y1 is 100. It's just straight up, right? 100. Okay, plus y2 is 30 cos 45. 30 cos 45 plus y3 is 80 sin 30. 80 sin 30. So this becomes, I'm just going to punch all this in in one shot in the calculator, because we're limited with space, right? We're limited with space. So we got, so what do we got? 30 cos 45. 30 times 45 trig cos plus 80 sin 30. Well, 80 sin of 30 is just 1 over 2, but I'm going to punch it anyway. Plus 80 times 30 second sin equals that, and then plus 100. So plus 100 is going to be 100 and 161.21. So that's the y direction, right? Which makes sense to a certain degree. Oh, what a mistake. What a mistake. Look at this. Look at this. This is going up. This is going up, but this is going down in vectors. Direction matters. So we're not adding 180 sin 30. We're subtracting 30 sin 80. Keep this in mind. Okay, so I'm going to punch this in again, right? How did I catch my own mistake? Well, I looked at this. I said if that's 100 and that's going to be this much, how could we go 168, which would be over here if we're going in this direction, right? It wouldn't be. So let's punch that in. This is where people get burned big time. So we're going to go 30 times 45 cos equals 21. It's the same thing as this, 21.21. And then we're going to subtract sin of 30 is opposite over that. It's 0.5 times 80, which is 40. So we're going to subtract 40 minus 40. We get negative because if you were just doing this and this, the direction is down, right? But then we're going to add 100. We're going to go up. So plus 100 plus 100 equals 81.21. 81.21. 81.21. Cool. We have the x and y components, right? Now, where are we going to draw this? Where are we going to draw this? Which part should I erase because we need more space? I can erase. I can erase. I'm going to take out these guys. Let's do this slowly. I'm going to take out these guys. So let's do the components on the original graph. Check this out. We had 90.49. Now, keep in mind, I went in this direction, so I called this positive, right? In physics, when you're doing vectors, it's up to you what you want to call positive, what you want to call negative. If you're going to stay with convention, I should have called this direction negative because that would have been positive. But I'm keeping track of it. It's the answer that matters, right? How I approach it is really up to me within a reason. You can't break the rules of mathematics, but you can define what you mean with whatever variables and directions and stuff, right? It's your game you're playing, right? So this is 90.49. So check this out. This was 80, so 90 would be around here. So the total measurement is going to be this way. Here, it's going to be a little bit less than that length if we're going to stay proportional, right? 90.49. So this is 90.49. And then we've got 81.21. Now remember, I called this down negative, up positive, and I subtracted this, so I know that's positive. So 80.21 is going to be here. I'm going to go all the way to here. So this is 80.21. So what you really have, you've got the legs, the components, the x and the y of your new vector, right? The total sum of all these three vectors added up. So what you can do is just take this guy and move it here to get a visual or take this guy and move it there. It's going to give you the same thing, right? So what I'm going to do is I'm going to draw this going up like this. So our total vector is going to look like this. So that's our total vector, right? There's two steps we need to do to finish this. We need to find the magnitude of this, and we need to find the angle of this. Well, the magnitude is easy, and the vector is easy. The magnitude is Pythagorean theorem. This is 80.21, 80.21, right? So all we're going to do a squared plus b squared equals c squared. This is the total. So a squared plus b squared equals c squared. Where should we do this? Let's do it over here. I'm going to erase these guys. Okay. So Pythagorean theorem says a squared plus b squared equals c squared. a squared plus b squared equals c squared. Where a is 90.49 squared plus 80.21 squared is equal to c squared. The total is c. Okay, we punch this in. And if you want to write this out, this becomes here 80.21. I want to write this out over here. We get more space. 80.21. So this becomes c squared is equal to, I'm just going to punch that in through the calculator. What's c squared? 90, 81.21 squared plus 90.49 squared, 90.49 squared equals 14,784, 14,784. Yep. And then to figure out what c is, you take the square root of both sides. So c is equal to squared of this duvaki. Squared of this duvaki is 121, 121.59. That's the magnitude here. Total is 121.59. Cool. We need to find the angle because we're doing vector. Vectors, right? We can't just give the scalar quantity because if we say, oh, the answer is 121.59, in which direction? We're talking vectors. So if we're going to do vectors, you go back to Sokotoa. Sokotoa, sine, cosine, and tangent, you have this length, you have this length, you have this length. You can use sine, cos, or tan. Let's use tan. We use sine and cos. Let's use tan. So tan of an angle, let's call this theta, tan of an angle, tan theta is equal to opposite, adjacent, what? Opposite over adjacent, right? Sokotoa, opposite over adjacent, opposite over adjacent, which is 80.21 over 90.49. Let's figure out what that is, right? 80.21, 80.21 divided by 90.49. Anyone want to guess what the angle is going to be? Let's see how accurate our drawing is. Check this out. I sometimes do this just for the fun of it. If this is 45 degrees, and if I drew this approximately proportionally, then this angle should be less than 45. So maybe it's around 38 degrees, 40 degrees. Let's check it out. Let's check it out. So you get, we get this, tan theta is equal to 0.0.886464. So theta is equal to tan inverse of this two, which is 0.8846. So theta is, let's do the inverse tan of it. 41.55 degrees, 41.55 degrees, and that's the angle. So this is 41.55 degrees. Which is pretty good, not bad. The accuracy was not bad. So your answer to this question would be, the solution to this would be magnitude would be 121. Oh, we should have used blue from the beginning. 121.59, and we would say, if we're going north and south, we could say 41.55 degrees north of west at 41.55 degrees north of west, or we could say 121.59 at, and figure out what this angle is, and you subtract from 90. So minus 90 is 48.44 at northwest, staying with this convention, oh, we erased it. Northwest 48.45, 48.45. That's vectors. You have to be careful where. You have to be careful in adding, subtracting in your directions. If you're going to call this positive, then everything this way is negative. If you want to call that positive, everything down is negative. We call this positive, and we call that positive because we're just working in that direction. I hope that helps. It's been a while since we did vectors, and this comes into play in physics a lot, a lot, especially in problems involving equilibrium and forces and magnitude and whatnot. Fun. Fun. That was great. Good math session. Good math session. Gang, I saw some follows and stuff flying through. I didn't cash the names, but thank you for the follows. Thank you for the support, gang. Also, multiplication matters, too, since it's not necessarily cumulative in linear algebra. Yeah, yeah, yeah. Multiply back. It changes a lot. I haven't done multiplication of vectors for a while. A while. I would have to look that stuff up. But adding and subtracting vectors? A lot. Bananas and chocolate chips. Hell, yes. I hope you have good snacks when you're doing mathematics. Not necessarily bananas and chocolates, but this is really something sweet.