0:05

Good, so, we're going to start out, we're not going to jump to rigid bodies,

Â we're going to do particles.

Â Hopefully, boring, okay, that means you know this stuff.

Â That's the idea.

Â So I got about 12 minutes.

Â Good, we'll get a good thing done.

Â What we want to talk about is the vector notation.

Â Some of this stuff is very, [LAUGH] If you go look at the field of attitude,

Â you have aerospace people, you have W people.

Â You've got an amazing number of nuclear physicist who some

Â reason dropped that field and went to attitude.

Â Markley and Landis are both two very famous rigid body attitude people through

Â all with nuclear physics background.

Â So everybody has their own notations, right?

Â And depending on which school, which books you learn from,

Â little different notations.

Â I want to set up what we're using in this class.

Â I think it's a very convenient notation,

Â but there's other ones you could think of, too.

Â Vector differentiation, very critical thing.

Â Momentum is a vector, position's are a vector, angular velocities are vectors.

Â They're all written with respect to rotating frames.

Â How do we fundamentally describe and do that?

Â That's what we're about.

Â And then basically, lots of brushing up on your own.

Â There's homework problems on this.

Â Solving chapter one, basic problem, can you get these things done?

Â And that's where you will see where you're a little.

Â And you'll go, I need to figure this one out.

Â This one was easy, this part I'm not quite sure.

Â So that's where you kind of, brings everybody up to the same page

Â before we really get into the 3D stuff, which would be about a week from now.

Â So, Vector Notation.

Â What is a vector?

Â What was your name?

Â >> Nick. >> Nick.

Â >> So a vector is essentially a direction of magnitude.

Â >> Direction of magnitude, perfect.

Â It's actually on the slide.

Â So if you printed out your slides, these are available online.

Â So you can get those down.

Â But that's really the fundamental of it.

Â A vector means, look, from me to that screen back there,

Â it's about 15 meters and it's straight ahead, right?

Â So if, Nick, if you were to apply that 15 meters and straight ahead,

Â would you get to the screen?

Â 2:07

>> No, I wouldn't right now.

Â >> Right. >> It'd be approximately that you were in

Â the wall.

Â >> Well, no, you'd be in the wall.

Â 15 meters, it's metric, I know, but still.

Â >> [LAUGH] >> I grew up in Switzerland.

Â I think in metric.

Â Inches, feet, elbows, it means nothing to me.

Â >> [LAUGH] >> Okay, so that's just a vector.

Â If you apply this vector straight ahead at 15 meters, it means different things for

Â different observers.

Â What does straight ahead mean?

Â It's a perfectly valid description.

Â You can say, where is the space station?

Â Well right now, it's about 6,000 kilometers that way.

Â 3:26

What's your name?

Â >> Maurice.

Â >> Maurice.

Â >> Yes.

Â >> Yes, it's a magnitude times a direction, magnitude times a direction.

Â We do all this.

Â There's different ways to write it, though.

Â Every vector can be written in infinity ways.

Â Because you can have one frame to write this vector, or

Â you could tweak that frame infinitesimally, and it's a new way.

Â Every little observer here will have a different orientation,

Â everybody's sitting slightly differently, it would be a slightly different

Â description, but it's fundamentally the same vector.

Â It goes 15 meters in some direction, and there's lots of ways to write it.

Â If you do it this way, this is now a 3 by 1 matrix.

Â 4:04

And I use a left superscript.

Â You can see e is my frame, my coordinate frame e.

Â And we have lots of frames, I really detest using x, y, and z.

Â As frame axes.

Â X's are typically coordinates.

Â And the axis is either your first, your second, your third.

Â And then you have six different frames.

Â With x, y, and z you can run out of letters in no time.

Â So if I have the e frame I tend to use e 1, 2, and 3.

Â Immediately I know what's my first, what's my second, what's my third.

Â This ordering will be critical when we do attitudes and

Â how these things all relate to each other.

Â And you know, but you could call them anything.

Â You can call them Bob, Harriet and Julia.

Â 4:42

It works, you know?

Â It's just names, so I used, if I have b frame, I have B 1, 2, 3.

Â If I have an e frame, I have e one, two, three.

Â It just makes my life easier, that's it.

Â But you have also r times.

Â Now is a different direction.

Â It's basically magnitude times the direction.

Â This is typically actually an orbit.

Â This is typically how we write our orbit problem.

Â That is to the center of the earth, your radius is 6,800 kilometers and

Â this is the direction to your satellite, right, from the center of the the earth.

Â It's a very convenient way to write it.

Â This one is a matrix notation.

Â 5:39

But not every 3 by 1 matrix is a vector.

Â And you will see both in this class.

Â For example, the direction cosine matrix, it's a 3 by 3.

Â It doesn't represent the nector.

Â It doesn't even represent the tensor, a multi-dimensional vector system.

Â It's just a grouping of numbers.

Â Sometimes we just have linear algebra.

Â It's math.

Â And these are just numbers, 4 and 3 and 2, and we've grouped them into a matrix so

Â we can use linear algebra methodologies to solve these equations, right?

Â If a matrix represents a vector, I typically write the left superscript.

Â 6:12

So this now says this 3 by 4 matrix, the first part, this is the magnitude.

Â And to get the vector you have to multiply times the first base vector,

Â which was e1, right?

Â So the e frame has the e1, e2, e3 based vector components.

Â And x times e1, y times e2, z times e3.

Â So this would be completely equivalent and correct.

Â If you just would write this, I don't know with respect to,

Â like we were talking earlier, you'd go 10, 0, 0.

Â Well, for me 10, 0, 0 is something different than for you 10, 0, 0.

Â It's different things, right?

Â If soon as you have a particular frame, it's very unique.

Â That comes in, okay?

Â So the key thing here is, every vector can be represented through a matrix but

Â not every matrix represents a vector and we'll be going this over and over again.

Â If you do prelims, this is a wonderful way to torture students.

Â 7:52

If you want to get a particular numerical answer, at some point, yes,

Â you have to get a frame.

Â But we will actually, in this class, the way I'm trying to teach you dynamics,

Â is I want you to forget about the frames at the beginning.

Â Just treat everything as vectors.

Â We're going to solve things in a vectorial way, because what you will get is

Â an answer, and it doesn't matter on the quarter frame.

Â The answer will literally be, well, it's about 7,000 kilometers that way, and

Â you go, okay, great.

Â Now, what does this mean?

Â The body that means that way was to his right, okay,

Â this was the orientation, and at the end, you can assign the frame.

Â If you assign a frame early you will have gazillions, and that's a technical term,

Â sines and co-sines everywhere, because now you're projecting everything into one

Â frame, and you have to make sure these sort of things work out.

Â I only do that if I absolutely have to,

Â otherwise everything is kept in a vectorial way.

Â This way you do it in a coordinate frame independent way.

Â So this is right, because you can say look here it's, oops let me get a pen.

Â This is one vector, this is another vector.

Â So have a, b, and

Â c ends up being a+b.

Â I didn't touch the corner frame but basically I can solve it.

Â You can, in planar 3D math this work as well.

Â You can write up vectors, you can add them, you can subtract them.

Â You can do other stuff and that's perfectly fine.

Â 9:42

>> It wouldn't be meaningful, right?

Â I mean Math Lab won't complain at you, it's just 1 3 by 1 plus another 3 by 1.

Â It gives you an answer.

Â It gave you what you asked.

Â It figured you knew what you're doing, of course it's silly Math Lab.

Â If it only knew, but this is not correct, all right?

Â So if you actually would numerically evaluate this,

Â this would come up with something that doesn't mean anything.

Â And so we really want to be careful that we have apples and apples.

Â If you add up matrix representations of vectors numerically,

Â all of the first components should be along the first axis, otherwise, you might

Â be adding these components with these components, and it makes no sense, right?

Â So this is very critical that you do this right.

Â But same thing here.

Â This is basically just a more compact way.

Â This is now actually bold for me means a vector, I don't use underscores.

Â In all the journals you see, the bold is basically a vector these days.

Â So bold is a vector, if I write it here, I typically have an underscore or

Â an arrow or something, because I can't do bold with my pen.

Â But that's an e-frame representation of that vector,

Â it's a 3 by 1 matrix equivalent to this.

Â Same thing here.

Â Same thing here.

Â This is also, of course,

Â would be wrong if you would actually numerically evaluate it.

Â Now halfway through the class I expect you guys to be big boys and girls.

Â We understand what's going on.

Â I may write these a few times but then it's implicitly understood

Â that if you numerically computed this you would be taking a b frame component and

Â map it into the e frame before you would add them up.

Â I'm just highlighting.

Â This is given in the b frame, this is an e frame.

Â I'm adding them, and I'm saying you guys know what you're doing, right?

Â You have to get everything into a common frame,

Â especially when we do the control development.

Â It just makes things quite a bit shorter.

Â So you will see this, as we get better and

Â better at this there's a lot of implied stuff that happens then, too.

Â Coordinate frames, hopefully very boring.

Â Coordinate frames you need 3 axis in 3D, given 3-dimensional space.

Â And the way I typically write the b frame, I have b, 1, 2, 3,

Â hats to me mean a unit vector.

Â Very simple, pretty common notation.

Â If there's an origin, I usually define it as 0 of b,

Â that's the origin of the b frame.

Â And you can fully define the frame this way.

Â It's important you have 3.

Â It's important that you label what's first, second, third.

Â That ordering will be critical.

Â You will see when we get to the rotation matrix and how this gets composed.

Â Otherwise, again, you end up with stuff in the wrong direction.

Â So you need to have a notation that tells you what's first, what's second,

Â what's third.

Â Right-handed means what?

Â What should you be able to do with these vectors,

Â if they satisfy a right-hand rule?

Â 12:15

>> Cross 2 is equal to 3?

Â >> Yeah, your first crossed your second, should be plus your third,

Â on your right-hand.

Â If you're confused on an exam, put an r on your right hand.

Â >> [LAUGH] >> I'm not joking, seriously.

Â There are many of you that are confused.

Â Exams are high stress periods, like, again, back to parachuting.

Â I can't tell you how many times I'm talking to somebody down on a walkie

Â talkie on a parachute, yeah, now, turn left, no, the other left.

Â Yeah, wait a minute, yeah.

Â 12:41

If you're confused, put an r on your hand.

Â Good, we got that.

Â In this class, typically, we're going to ignore the origin.

Â That's really the orbits part that's on that N 50/50.

Â We will typically have a short hand that we just say look, the b frame is b1, 2, 3.

Â because all we're caring about is the attitude.

Â So that's just notationally where things go.

Â Okay, it's 9:15, good, right on time.

Â So next time we're going to start up with angular velocity which is what we need.

Â What does this mean?

Â And this delves into taking derivatives as a rotating frame which requires angular

Â velocity.

Â So, good, thanks everybody.

Â Grab the homeworks and we'll see you on Thursday.

Â