Good, so, we're going to start out, we're not going to jump to rigid bodies, we're going to do particles. Hopefully, boring, okay, that means you know this stuff. That's the idea. So I got about 12 minutes. Good, we'll get a good thing done. What we want to talk about is the vector notation. Some of this stuff is very, [LAUGH] If you go look at the field of attitude, you have aerospace people, you have W people. You've got an amazing number of nuclear physicist who some reason dropped that field and went to attitude. Markley and Landis are both two very famous rigid body attitude people through all with nuclear physics background. So everybody has their own notations, right? And depending on which school, which books you learn from, little different notations. I want to set up what we're using in this class. I think it's a very convenient notation, but there's other ones you could think of, too. Vector differentiation, very critical thing. Momentum is a vector, position's are a vector, angular velocities are vectors. They're all written with respect to rotating frames. How do we fundamentally describe and do that? That's what we're about. And then basically, lots of brushing up on your own. There's homework problems on this. Solving chapter one, basic problem, can you get these things done? And that's where you will see where you're a little. And you'll go, I need to figure this one out. This one was easy, this part I'm not quite sure. So that's where you kind of, brings everybody up to the same page before we really get into the 3D stuff, which would be about a week from now. So, Vector Notation. What is a vector? What was your name? >> Nick. >> Nick. >> So a vector is essentially a direction of magnitude. >> Direction of magnitude, perfect. It's actually on the slide. So if you printed out your slides, these are available online. So you can get those down. But that's really the fundamental of it. A vector means, look, from me to that screen back there, it's about 15 meters and it's straight ahead, right? So if, Nick, if you were to apply that 15 meters and straight ahead, would you get to the screen? >> No, I wouldn't right now. >> Right. >> It'd be approximately that you were in the wall. >> Well, no, you'd be in the wall. 15 meters, it's metric, I know, but still. >> [LAUGH] >> I grew up in Switzerland. I think in metric. Inches, feet, elbows, it means nothing to me. >> [LAUGH] >> Okay, so that's just a vector. If you apply this vector straight ahead at 15 meters, it means different things for different observers. What does straight ahead mean? It's a perfectly valid description. You can say, where is the space station? Well right now, it's about 6,000 kilometers that way. That's it. Now to actually do something with that, you need more information. Like, who said that, what was to my right? And which way were they facing, what was their orientation? You need to have all of those components. But a vector fundamentally just is a direction and a magnitude. If we write a vector out as saying r as the vector is x times e 1, y times e 2, z times e 3. And you can see e 1 2 3 are a right-handed coordinate system. Unit vectors. Unit vectors it means the vector has length of 1, all right? And you basically break down the dot products, you get your x, your y, your z components. This is a way we can write this. Is this right-hand side still a vector? What's your name? >> Maurice. >> Maurice. >> Yes. >> Yes, it's a magnitude times a direction, magnitude times a direction. We do all this. There's different ways to write it, though. Every vector can be written in infinity ways. Because you can have one frame to write this vector, or you could tweak that frame infinitesimally, and it's a new way. Every little observer here will have a different orientation, everybody's sitting slightly differently, it would be a slightly different description, but it's fundamentally the same vector. It goes 15 meters in some direction, and there's lots of ways to write it. If you do it this way, this is now a 3 by 1 matrix. And I use a left superscript. You can see e is my frame, my coordinate frame e. And we have lots of frames, I really detest using x, y, and z. As frame axes. X's are typically coordinates. And the axis is either your first, your second, your third. And then you have six different frames. With x, y, and z you can run out of letters in no time. So if I have the e frame I tend to use e 1, 2, and 3. Immediately I know what's my first, what's my second, what's my third. This ordering will be critical when we do attitudes and how these things all relate to each other. And you know, but you could call them anything. You can call them Bob, Harriet and Julia. It works, you know? It's just names, so I used, if I have b frame, I have B 1, 2, 3. If I have an e frame, I have e one, two, three. It just makes my life easier, that's it. But you have also r times. Now is a different direction. It's basically magnitude times the direction. This is typically actually an orbit. This is typically how we write our orbit problem. That is to the center of the earth, your radius is 6,800 kilometers and this is the direction to your satellite, right, from the center of the the earth. It's a very convenient way to write it. This one is a matrix notation. A matrix is not a vector. Math Lab will argue with you because in Math Lab you put everything in as 3 by 1's, right? But what you're doing fundamentally is you're taking a vector, you're choosing a particular coordinate frame, and you're breaking down the first, second or third components. And now this 3 by 1 can represent a vector. But not every 3 by 1 matrix is a vector. And you will see both in this class. For example, the direction cosine matrix, it's a 3 by 3. It doesn't represent the nector. It doesn't even represent the tensor, a multi-dimensional vector system. It's just a grouping of numbers. Sometimes we just have linear algebra. It's math. And these are just numbers, 4 and 3 and 2, and we've grouped them into a matrix so we can use linear algebra methodologies to solve these equations, right? If a matrix represents a vector, I typically write the left superscript. So this now says this 3 by 4 matrix, the first part, this is the magnitude. And to get the vector you have to multiply times the first base vector, which was e1, right? So the e frame has the e1, e2, e3 based vector components. And x times e1, y times e2, z times e3. So this would be completely equivalent and correct. If you just would write this, I don't know with respect to, like we were talking earlier, you'd go 10, 0, 0. Well, for me 10, 0, 0 is something different than for you 10, 0, 0. It's different things, right? If soon as you have a particular frame, it's very unique. That comes in, okay? So the key thing here is, every vector can be represented through a matrix but not every matrix represents a vector and we'll be going this over and over again. If you do prelims, this is a wonderful way to torture students. So vector additions. If I write simply this vector is equal to this vector plus this vector. Sorry, what's your name? >> Matt. >> Matt. Correct, not correct? >> If they're all in the same frame, I think it would work. >> Do you have to write q and r and p in the same frame for this math to work? >> No, you just wouldn't do it on the same way. >> [SOUND]. >> I guess you can have all three vectors no matter what, but the way you do it depends on the frames. >> You should've stopped at the first part. >> [LAUGH]. >> Vectors fundamentally, I can add. I can subtract. I can do dock products. I can do cross products. I can do all kinds of vector math without ever assigning a frame. If you want to get a particular numerical answer, at some point, yes, you have to get a frame. But we will actually, in this class, the way I'm trying to teach you dynamics, is I want you to forget about the frames at the beginning. Just treat everything as vectors. We're going to solve things in a vectorial way, because what you will get is an answer, and it doesn't matter on the quarter frame. The answer will literally be, well, it's about 7,000 kilometers that way, and you go, okay, great. Now, what does this mean? The body that means that way was to his right, okay, this was the orientation, and at the end, you can assign the frame. If you assign a frame early you will have gazillions, and that's a technical term, sines and co-sines everywhere, because now you're projecting everything into one frame, and you have to make sure these sort of things work out. I only do that if I absolutely have to, otherwise everything is kept in a vectorial way. This way you do it in a coordinate frame independent way. So this is right, because you can say look here it's, oops let me get a pen. This is one vector, this is another vector. So have a, b, and c ends up being a+b. I didn't touch the corner frame but basically I can solve it. You can, in planar 3D math this work as well. You can write up vectors, you can add them, you can subtract them. You can do other stuff and that's perfectly fine. But let's go back to the other. So this is perfectly fine. Here if I have matrix representations of vectors. And this r vector's broken up into e frame, this p vector's broken up into b frame. And you just add up the first components, the second components, and third components. Is that going to work? Is it Andre? >> Yeah. >> What do you think? >> It would work, but it wouldn't be meaningful. >> It wouldn't be meaningful, right? I mean Math Lab won't complain at you, it's just 1 3 by 1 plus another 3 by 1. It gives you an answer. It gave you what you asked. It figured you knew what you're doing, of course it's silly Math Lab. If it only knew, but this is not correct, all right? So if you actually would numerically evaluate this, this would come up with something that doesn't mean anything. And so we really want to be careful that we have apples and apples. If you add up matrix representations of vectors numerically, all of the first components should be along the first axis, otherwise, you might be adding these components with these components, and it makes no sense, right? So this is very critical that you do this right. But same thing here. This is basically just a more compact way. This is now actually bold for me means a vector, I don't use underscores. In all the journals you see, the bold is basically a vector these days. So bold is a vector, if I write it here, I typically have an underscore or an arrow or something, because I can't do bold with my pen. But that's an e-frame representation of that vector, it's a 3 by 1 matrix equivalent to this. Same thing here. Same thing here. This is also, of course, would be wrong if you would actually numerically evaluate it. Now halfway through the class I expect you guys to be big boys and girls. We understand what's going on. I may write these a few times but then it's implicitly understood that if you numerically computed this you would be taking a b frame component and map it into the e frame before you would add them up. I'm just highlighting. This is given in the b frame, this is an e frame. I'm adding them, and I'm saying you guys know what you're doing, right? You have to get everything into a common frame, especially when we do the control development. It just makes things quite a bit shorter. So you will see this, as we get better and better at this there's a lot of implied stuff that happens then, too. Coordinate frames, hopefully very boring. Coordinate frames you need 3 axis in 3D, given 3-dimensional space. And the way I typically write the b frame, I have b, 1, 2, 3, hats to me mean a unit vector. Very simple, pretty common notation. If there's an origin, I usually define it as 0 of b, that's the origin of the b frame. And you can fully define the frame this way. It's important you have 3. It's important that you label what's first, second, third. That ordering will be critical. You will see when we get to the rotation matrix and how this gets composed. Otherwise, again, you end up with stuff in the wrong direction. So you need to have a notation that tells you what's first, what's second, what's third. Right-handed means what? What should you be able to do with these vectors, if they satisfy a right-hand rule? What's the easy mathematical trick? >> Cross products. >> Cross products, okay. And which do you have to take cross products on? >> Cross 2 is equal to 3? >> Yeah, your first crossed your second, should be plus your third, on your right-hand. If you're confused on an exam, put an r on your right hand. >> [LAUGH] >> I'm not joking, seriously. There are many of you that are confused. Exams are high stress periods, like, again, back to parachuting. I can't tell you how many times I'm talking to somebody down on a walkie talkie on a parachute, yeah, now, turn left, no, the other left. Yeah, wait a minute, yeah. If you're confused, put an r on your hand. Good, we got that. In this class, typically, we're going to ignore the origin. That's really the orbits part that's on that N 50/50. We will typically have a short hand that we just say look, the b frame is b1, 2, 3. because all we're caring about is the attitude. So that's just notationally where things go. Okay, it's 9:15, good, right on time. So next time we're going to start up with angular velocity which is what we need. What does this mean? And this delves into taking derivatives as a rotating frame which requires angular velocity. So, good, thanks everybody. Grab the homeworks and we'll see you on Thursday.