Little bit of a review. I'm not going to quiz you on the syllabus, that's boring, even for me. We did a little bit of technical stuff, and we're going to start using this now. The material we're covering today, it's very basic, you're doing a bunch of homework kind of applying this stuff. But it's the kind of tool that's really important, because we'll be using it throughout the class. And these are the kind of things that people trip up over, over, and over, and over again. So the concepts may seem simple, the applications can quickly become tricky, and wait a minute, and what happens in this case, and what in this case, and that's what you will see throughout the course. And it all has to do with vectors and matrices. And today, we'll get into differentiating vectors, what does that actually mean, and we'll go from there. So let's see, Sheila, what is a vector? >> It's something that describes the magnitude and direction. >> Basically, yeah, just think arrow, right, that's a vector. It has a certain length and it has a certain direction. So is it possible to write your mathematics in vectorial form without ever specifying a frame? Chuck? >> You could, but it would change based on. The example that you gave last week when you were like the wall is 15 meters- >> It was this week, but last lecture, but that's okay. [LAUGH] >> That's different, because [INAUDIBLE]. >> Yeah, right. >> [INAUDIBLE] >> But fundamentally, we talked about, let me just talk about position vectors. Most people are very comfortable with positions. And the same things will apply when we talk about omegas, angular velocities, or angular momentum vectors, they're just vectorial things. But people are quite comfortable with positions, I go from here to here, from there to there. And everyone of those descriptions is actually a vectorial form. So if you have this, let's say I'm doing an orbit, then I might have a certain radius times a direction, right? This is fully defined, you can do your math and you can solve for things this way. In fact, when you take the derivatives, we will see how we do that. You take the inertial derivative of this, you just get your orbit rate + r times an angular rate, think of it as like a central coordinate system. You're going to get these things, and we've broken down the velocity now into a radial component and tangential component. What does radial actually mean? Well, now you need to know, where is that frame, right,? But we've chosen here a rotating frame, and it actually makes life, in many cases, much simpler. For the attitude problem, we tend to put everything into the body frame, and you will see later why. Now, but everything's in a body frame, and so, in the body, the body tumbles, it moves, it rotates. We just have to, fundamentally, tackle how do we differentiate, as seen by body frames or inertial frames, and dealing with that. But that was a basic vector, what if I want to write this vector? So let me say, I want to define this orbit frame. These are coordinates I didn't use last time, so I'm trying to mix things up a little bit. But let's just have a coordinate frame for the orbit. What all do we have to define? I'm sorry, what was your name? >> Manor. >> So Manor, what all do you need to specify, if you want to specify an orbit frame, or any coordinate frame? >> The three Xs. >> Okay, we will need three Xs, anything else? >> The origin. >> If you want a full thing, you might define the origin, or you might say, look there's a point A, you're tracking astronaut A in space, right, that's it. And then, you needed three vectors. What type of vectors do we need here? Is it Nicholas? No. >> Charles. >> Charles, that's right. The chalk and Charles, I need to remember that. Charles, what's special about these vectors? >> That they have a defining name. >> Yes, they're specifying, so Manor's saying, you need three vectors to specify a coordinate frame. Could I put in any vectors? >> No, we need them to be orthogonal. >> Okay, they need to be orthogonal, good, what else? What was your name? >> Ben. >> I'm sorry? >> Ben. >> Ben, okay, what else? >> Unit vectors? >> Okay, unit vectors, what else? >> The start of the origin. >> No. >> No. >> No, unit vectors are really direction vectors, so as direction vectors, I'm not specifying this vector I've drawn is a magnitude times a direction. I could shift it, and it's the same length and the same direction, it's the same vector. It may start from a different point, but as a direction vector, and that's what goes into here, we need three direction vectors. Where it starts from doesn't matter, and actually that's a good question that comes up when we make drawings of these frames, because everybody tries to draw them right at the origin, and then that origin becomes so crowded. You need a magnifying glass to be able to distinguish stuff. You can actually just shift them outward and say, okay, that's the direction. So think of East, West and North, for all of us, let's pretend this wall is East, because I don't know, actually, how we're oriented in this university. I should know after 10 years, but let's say this wall is East. East is the same direction for all of us, but you're sitting in a different location than I, but the East direction is still the same, right? So as a direction vector, where you start from doesn't really matter, you're just saying look, I'm going to move five meters North, six meters East, five meters North, again, and it will eventually get you to the place, if you start, of course, in the right location. And that's when we get back to position descriptions. For unit vectors, the origin doesn't actually matter, it's just which way are you heading, okay, so very glad you brought that up. So we need to have three vectors here, orthogonal. What does orthogonal actually mean? >> When you do a cross product [INAUDIBLE], cross product of two of them produces the [INAUDIBLE] in some direction? >> That would work, that would work. It's not the definition I was quite thinking of, anybody have a different definition? >> [INAUDIBLE] product to be zero. >> Yes, so if you do a dot product right, here's one vector, here's another vector. If they're orthogonal, then the projection, the dot product, essentially, geometric is projection. The projection of this vector onto this vector is going to be zero. The cross product thing would work too, because you do the cross product of one times the other, and it has to give you precisely the third, because these are unit vectors, right? So if you look at the cross product rule, it's the magnitude of one, magnitude of the other. Then you have the sine of the angle between them, which is 90 degrees, so sine of 90 is 1, off you go, right? That would work too, actually, yep. So orthogonal means right-handed, even in three dimensional space. There's one thing we're missing here on these vectors. >> [INAUDIBLE] >> That's basically orthogonal, that would be an equivalent description. Yep, that means they are at right angles. Okay, so yep, we've got that. Yes, sir, what was your name? >> Bryan. >> Bryan, thank you. >> Do the order [INAUDIBLE]. >> The order, that's something that's often overlooked. So the order is important, so let me just ask you a simple question. I'm going to have an O frame, I am going to call this one, O2, O3, and O1. So Brian, is that correct? >> Yes. >> Because? >> because the order of the three is [INAUDIBLE] a little bit. I'm not sure how to say it, but two cross three is [INAUDIBLE]. >> Okay, so with the ordering it's back to the cross product stuff, right? That the first one crossed with the second has to give you the third, and then you can shift that permutation. Second crossed third has to give you the first, third crossed first has to give you the second, right? And you're saying that is true? >> I think so. [LAUGH] >> So let me draw a figure first. So these two are kind of pointing out towards us, and this third one just points up, right? So if I declare this as O1, this as O2, and this as O3. Would this be a correct ordering? Sorry, what's your name? White shirt. >> Kevin. >> I'm sorry? >> Kevin. >> Keaven? >> Kevin, K-E-V-I-N. >> Kevin. >> Kevin. >> Kevin, Kevin, thank you. I do actually have a hearing aid on my left ear, so some frequencies I don't hear very well. So if I seem hard of hearing, I actually am. Yes, so you were shaking your head, would this be a correct definition for this frame? >> If you cross O2 with O3, then you get minus O1 in the [INAUDIBLE]. >> So what made Bryan think this was a correct description? The labeling, the numbers, right? We tend to write, I do, certainly, I would define this as my first axis, second axis, and third axis. But, in fact, here I've done something that's saying is a big no, no in this class, don't write a left-handed coordinate frame, right? You want to write always right-handed coordinate frames. But you could label them differently, I could go here and say this one, I'm going to switch these and say, okay, this one is 2 and then this one is 1, in which case this is a correct frame, right? So the naming of these axis is really arbitrary. Whatever you call them, you can call them anything you wish. Typically, we don't quite write them this way, because this gets a little confusing, right? Typically, what we would do is we would have a B frame. In fact, I dropped the origin, because in this class we're doing all attitude problems, we don't typically. The origins you will see, we take into account when we write our positions. We're going from here to here to here, and that's accounted for there, the rest we don't need. But I would typically write them this way. [COUGH] And this has to be a right-handed frame, so b1, b2, b3 have to be orthogonal unit length, orthogonal, as we were talking about, and sorry, perpendicular. But the first one crossed the second one, has to be the third, if that doesn't match with your graphic from your doodle that you made of the problem statement, then something's off, right? But what we sometimes do too is maybe b2 is shared with the e frame, and now there might be a e theta 1 axis. And you find instead of having b2 and e theta rotating around, then always remembering b2 is equal to e theta, you might choose to rewrite this and say, well, this is b1, e theta, b3. And this can be helpful too, because then we when you look at the definitions, you quickly realize, well, the first crossed second is equal, has to be the third. I know that e theta has to be orthogonal to b1 and b3, right? So in the end, these frames are just names, what names do you put in? I always suggest be lazy in this class, as a dynamicist being lazy is a big virtue. That means you're seeking the easiest path to get from the problem statement to the kinematic, dynamic, kinetic descriptions, all right? And sometimes you have lots of frames, so you really want to have simplicity, whatever makes your algebra easier. Typically, if I have an e frame, I'm calling it e1,2,3, and then axis are hopefully drawn to make it right-handed and work. If I have a b frame, it's b1,2,3, if have an e frame that relates to a b frame, I have to look at the problem statement, sometimes I keep them separate, sometimes I put them together. And you will see in this homework number one, you'll be going through different problems, and there's different ways you can solve it, but see what works well for you, okay? But this is all just basics of coordinate frames, and vectors, and stuff. Good, okay, any questions on frames, vectors? I have one for you. Is this a vector, Andre? >> Absolutely. >> You say it with great conviction, I love that. Unfortunately, the answer is completely bogus, but kudos for saying it with great convictions. Why is that bogus? What was your name? >> Cody. >> Cody? >> Yep. [INAUDIBLE] >> What would we need to turn this into a vector? [SOUND] If I'm saying look, I actually before you guys showed up, hid a $5 bill in this room, and the location is 5, 3 and 2. Before you go hunting, there's no $5 bill in here, sorry. But let's pretend there's a $5 bill somewhere, and it tells you it's at 5 meters, 3 meters, and 2 meters. Andre, would you know where in the room this is, even if the origin was this corner, I'm telling you that much? >> You'd need to define a coordinate system, first. >> Right, you have to know 5 meters in what direction, right? This just gives you the magnitudes, but not the directions. So if we want to turn this into, a matrix can represent a vector. So how do we make this 3 by 1 matrix now represent a vector? What do we have to change? I'm sorry, yeah, what was your name? >> Evan. >> Evan, thank you. >> Just specify the frame as like a- >> And how do we do this? >> A left superscript. >> A left superscript. So if we say e, then this is equivalent to (5 m) e1 + (3 m) e2 + (2 m) e3, right? Now, you've specified from the origin, which you have to know where e1, 2, and 3 point, but you know in what directions to move to get to the next point. And this is a distinction that often people, not just students, I see a lot of researchers and journal papers who are giving reviews, and I'm like, no, this doesn't make sense. In Matlab, you just treated this way, but there's an implied thing in Matlab is that when you're adding some number, you have 1, 2, 3 in Matlab plus 4, 5, 6, and you add these things up. If these things represent vectors, what do you have to make sure of? What's your name again? >> Matt. >> Matt, thank you. >> That they're in the same frame. >> Right, and many problems, freshmen dynamics, sophomore dynamics, we just put everything back into inertial frame. That's why you have to learn all those trig identities, because quickly there's logs and sines and cosines, and you add things up and put it together and there's a magic trig identity that simplifies it. And in the end, it's 4 meters in the radial direction. You're going to see a much easier way to get to it here, so we just have to make sure they're both in the e frame, then the answer will make sense. But you have to interpret it as an e frame, if you need it in a different frame, you have to do coordinate transformations. How do we map vector components from one frame into another frame? This is something I know most of you have all seen before. What do we use to do a coordinate frame conversion, what do we have to do? >> Rotation matrices. >> Rotation matrices, direction cosine, the transformation, whatever you want to call it, different name. And we'll get to that in much more detail, and you will see all of the intricate connections of this stuff. So good, that's a distinction, this by itself is just a matrix. This like this is a matrix that represents a vector, so every vector can be represented in matrix form. You break down vector components along three orthogonal axes, but not every matrix represents a vector. In this class, you will see sometimes things that we write this way without this left superscript, and it's literally just a three by one stack of numbers. We had three equations and it was easiest, again laziness, it was easiest to write in a matrix form, because then we use classic matrix, linear algebra math tools, and solve for this stuff, all right, so we see both. Good, now, moving on to just a quick review of basic coordinate frames, vectors, and the difference between matrices and vectors.