0:05

Little bit of a review.

Â I'm not going to quiz you on the syllabus, that's boring, even for me.

Â We did a little bit of technical stuff, and we're going to start using this now.

Â The material we're covering today, it's very basic,

Â you're doing a bunch of homework kind of applying this stuff.

Â But it's the kind of tool that's really important,

Â because we'll be using it throughout the class.

Â And these are the kind of things that people trip up over, over, and over, and

Â over again.

Â So the concepts may seem simple, the applications can quickly become tricky,

Â and wait a minute, and what happens in this case, and what in this case, and

Â that's what you will see throughout the course.

Â And it all has to do with vectors and matrices.

Â And today, we'll get into differentiating vectors, what does that actually mean, and

Â we'll go from there.

Â So let's see, Sheila,

Â what is a vector?

Â >> It's something that describes the magnitude and direction.

Â >> Basically, yeah, just think arrow, right, that's a vector.

Â It has a certain length and it has a certain direction.

Â So is it possible to write your mathematics in vectorial form

Â without ever specifying a frame?

Â Chuck?

Â >> You could, but it would change based on.

Â 1:28

The example that you gave last week when you were like the wall is 15 meters-

Â >> It was this week, but last lecture, but

Â that's okay.

Â [LAUGH] >> That's different, because [INAUDIBLE].

Â >> Yeah, right.

Â >> [INAUDIBLE] >> But fundamentally, we talked about,

Â let me just talk about position vectors.

Â Most people are very comfortable with positions.

Â And the same things will apply when we talk about omegas, angular velocities,

Â or angular momentum vectors, they're just vectorial things.

Â But people are quite comfortable with positions, I go from here to here,

Â from there to there.

Â And everyone of those descriptions is actually a vectorial form.

Â So if you have this, let's say I'm doing an orbit, then I might have a certain

Â radius times a direction, right?

Â 2:09

This is fully defined, you can do your math and you can solve for

Â things this way.

Â In fact, when you take the derivatives, we will see how we do that.

Â You take the inertial derivative of this, you just get your orbit rate +

Â r times an angular rate, think of it as like a central coordinate system.

Â You're going to get these things, and

Â we've broken down the velocity now into a radial component and tangential component.

Â What does radial actually mean?

Â Well, now you need to know, where is that frame, right,?

Â But we've chosen here a rotating frame, and it actually makes life, in many cases,

Â much simpler.

Â For the attitude problem, we tend to put everything into the body frame, and

Â you will see later why.

Â Now, but everything's in a body frame, and so, in the body,

Â the body tumbles, it moves, it rotates.

Â We just have to, fundamentally, tackle how do we differentiate,

Â as seen by body frames or inertial frames, and dealing with that.

Â But that was a basic vector, what if I want to write this vector?

Â So let me say, I want to define this orbit frame.

Â These are coordinates I didn't use last time, so

Â I'm trying to mix things up a little bit.

Â But let's just have a coordinate frame for the orbit.

Â What all do we have to define?

Â I'm sorry, what was your name?

Â >> Manor.

Â >> So Manor, what all do you need to specify,

Â if you want to specify an orbit frame, or any coordinate frame?

Â >> The three Xs.

Â >> Okay, we will need three Xs, anything else?

Â >> The origin.

Â >> If you want a full thing, you might define the origin, or you might say,

Â look there's a point A, you're tracking astronaut A in space, right, that's it.

Â And then, you needed three vectors.

Â 3:54

What type of vectors do we need here?

Â Is it Nicholas?

Â No.

Â >> Charles.

Â >> Charles, that's right.

Â The chalk and Charles, I need to remember that.

Â Charles, what's special about these vectors?

Â >> That they have a defining name.

Â >> Yes, they're specifying, so Manor's saying,

Â you need three vectors to specify a coordinate frame.

Â Could I put in any vectors?

Â >> No, we need them to be orthogonal.

Â >> Okay, they need to be orthogonal, good, what else?

Â What was your name?

Â >> Ben.

Â >> I'm sorry?

Â >> Ben.

Â >> Ben, okay, what else?

Â >> Unit vectors?

Â >> Okay, unit vectors, what else?

Â >> The start of the origin.

Â >> No.

Â >> No.

Â >> No, unit vectors are really direction vectors, so as direction vectors,

Â I'm not specifying this vector I've drawn is a magnitude times a direction.

Â I could shift it, and it's the same length and

Â the same direction, it's the same vector.

Â It may start from a different point, but as a direction vector, and

Â that's what goes into here, we need three direction vectors.

Â Where it starts from doesn't matter, and actually that's a good question that comes

Â up when we make drawings of these frames, because everybody tries to draw them right

Â at the origin, and then that origin becomes so crowded.

Â You need a magnifying glass to be able to distinguish stuff.

Â You can actually just shift them outward and say, okay, that's the direction.

Â So think of East, West and North, for all of us, let's pretend this wall is East,

Â because I don't know, actually, how we're oriented in this university.

Â I should know after 10 years, but let's say this wall is East.

Â East is the same direction for all of us, but you're sitting in a different

Â location than I, but the East direction is still the same, right?

Â So as a direction vector, where you start from doesn't really matter,

Â you're just saying look, I'm going to move five meters North, six meters East,

Â five meters North, again, and it will eventually get you to the place,

Â if you start, of course, in the right location.

Â And that's when we get back to position descriptions.

Â For unit vectors, the origin doesn't actually matter,

Â it's just which way are you heading, okay, so very glad you brought that up.

Â So we need to have three vectors here, orthogonal.

Â What does orthogonal actually mean?

Â >> When you do a cross product [INAUDIBLE],

Â cross product of two of them produces the [INAUDIBLE] in some direction?

Â >> That would work, that would work.

Â It's not the definition I was quite thinking of,

Â anybody have a different definition?

Â >> [INAUDIBLE] product to be zero.

Â 6:27

>> Yes, so if you do a dot product right, here's one vector, here's another vector.

Â If they're orthogonal, then the projection, the dot product,

Â essentially, geometric is projection.

Â The projection of this vector onto this vector is going to be zero.

Â The cross product thing would work too,

Â because you do the cross product of one times the other, and

Â it has to give you precisely the third, because these are unit vectors, right?

Â So if you look at the cross product rule, it's the magnitude of one,

Â magnitude of the other.

Â Then you have the sine of the angle between them, which is 90 degrees,

Â so sine of 90 is 1, off you go, right?

Â That would work too, actually, yep.

Â So orthogonal means right-handed, even in three dimensional space.

Â 7:04

There's one thing we're missing here on these vectors.

Â >> [INAUDIBLE] >> That's basically orthogonal,

Â that would be an equivalent description.

Â Yep, that means they are at right angles.

Â Okay, so yep, we've got that.

Â Yes, sir, what was your name?

Â >> Bryan. >> Bryan, thank you.

Â >> Do the order [INAUDIBLE].

Â >> The order, that's something that's often overlooked.

Â So the order is important, so let me just ask you a simple question.

Â I'm going to have an O frame,

Â I am going to call this one, O2, O3, and O1.

Â So Brian, is that correct?

Â 9:30

The labeling, the numbers, right?

Â We tend to write, I do, certainly, I would define this as my first axis,

Â second axis, and third axis.

Â But, in fact, here I've done something that's saying is a big no,

Â no in this class, don't write a left-handed coordinate frame, right?

Â You want to write always right-handed coordinate frames.

Â But you could label them differently, I could go here and

Â say this one, I'm going to switch these and say,

Â okay, this one is 2 and then this one is 1,

Â in which case this is a correct frame, right?

Â So the naming of these axis is really arbitrary.

Â Whatever you call them, you can call them anything you wish.

Â Typically, we don't quite write them this way,

Â because this gets a little confusing, right?

Â Typically, what we would do is we would have a B frame.

Â In fact, I dropped the origin,

Â because in this class we're doing all attitude problems, we don't typically.

Â The origins you will see, we take into account when we write our positions.

Â We're going from here to here to here, and that's accounted for there,

Â the rest we don't need.

Â But I would typically write them this way.

Â [COUGH] And this has to be a right-handed frame,

Â so b1, b2, b3 have to be orthogonal unit length, orthogonal,

Â as we were talking about, and sorry, perpendicular.

Â But the first one crossed the second one,

Â has to be the third, if that doesn't match with your graphic from your

Â doodle that you made of the problem statement, then something's off, right?

Â But what we sometimes do too is maybe b2 is shared with the e frame,

Â and now there might be a e theta 1 axis.

Â And you find instead of having b2 and e theta rotating around, then always

Â remembering b2 is equal to e theta, you might choose to rewrite this and say,

Â well, this is b1, e theta, b3.

Â And this can be helpful too, because then we when you look at the definitions, you

Â quickly realize, well, the first crossed second is equal, has to be the third.

Â I know that e theta has to be orthogonal to b1 and b3, right?

Â So in the end, these frames are just names, what names do you put in?

Â I always suggest be lazy in this class, as a dynamicist being lazy is a big virtue.

Â That means you're seeking the easiest path to get from the problem statement

Â to the kinematic, dynamic, kinetic descriptions, all right?

Â And sometimes you have lots of frames,

Â so you really want to have simplicity, whatever makes your algebra easier.

Â Typically, if I have an e frame, I'm calling it e1,2,3,

Â and then axis are hopefully drawn to make it right-handed and work.

Â If I have a b frame, it's b1,2,3, if have an e frame that relates to a b frame,

Â I have to look at the problem statement, sometimes I keep them separate,

Â sometimes I put them together.

Â And you will see in this homework number one,

Â you'll be going through different problems, and

Â there's different ways you can solve it, but see what works well for you, okay?

Â But this is all just basics of coordinate frames, and vectors, and stuff.

Â 13:05

[INAUDIBLE] >> What

Â would we need to turn this into a vector?

Â [SOUND] If I'm saying look, I actually before you guys showed up,

Â hid a $5 bill in this room, and the location is 5, 3 and 2.

Â Before you go hunting, there's no $5 bill in here, sorry.

Â But let's pretend there's a $5 bill somewhere, and

Â it tells you it's at 5 meters, 3 meters, and 2 meters.

Â Andre, would you know where in the room this is,

Â even if the origin was this corner, I'm telling you that much?

Â >> You'd need to define a coordinate system, first.

Â >> Right, you have to know 5 meters in what direction, right?

Â This just gives you the magnitudes, but not the directions.

Â So if we want to turn this into, a matrix can represent a vector.

Â 13:53

So how do we make this 3 by 1 matrix now represent a vector?

Â What do we have to change?

Â I'm sorry, yeah, what was your name?

Â >> Evan. >> Evan, thank you.

Â >> Just specify the frame as like a- >> And how do we do this?

Â >> A left superscript.

Â >> A left superscript.

Â So if we say e, then this is

Â equivalent to (5 m) e1 + (3

Â m) e2 + (2 m) e3, right?

Â Now, you've specified from the origin, which you have to know where e1, 2, and

Â 3 point, but you know in what directions to move to get to the next point.

Â And this is a distinction that often people, not just students,

Â I see a lot of researchers and

Â journal papers who are giving reviews, and I'm like, no, this doesn't make sense.

Â 14:41

In Matlab, you just treated this way, but there's an implied thing in Matlab is that

Â when you're adding some number, you have 1, 2, 3 in Matlab plus 4,

Â 5, 6, and you add these things up.

Â If these things represent vectors, what do you have to make sure of?

Â What's your name again?

Â >> Matt. >> Matt, thank you.

Â >> That they're in the same frame.

Â >> Right, and many problems, freshmen dynamics, sophomore dynamics,

Â we just put everything back into inertial frame.

Â That's why you have to learn all those trig identities,

Â because quickly there's logs and sines and cosines, and you add things up and

Â put it together and there's a magic trig identity that simplifies it.

Â And in the end, it's 4 meters in the radial direction.

Â You're going to see a much easier way to get to it here, so we just have to

Â make sure they're both in the e frame, then the answer will make sense.

Â But you have to interpret it as an e frame,

Â if you need it in a different frame, you have to do coordinate transformations.

Â How do we map vector components from one frame into another frame?

Â This is something I know most of you have all seen before.

Â 15:45

What do we use to do a coordinate frame conversion, what do we have to do?

Â >> Rotation matrices.

Â >> Rotation matrices, direction cosine, the transformation,

Â whatever you want to call it, different name.

Â And we'll get to that in much more detail,

Â and you will see all of the intricate connections of this stuff.

Â So good, that's a distinction, this by itself is just a matrix.

Â This like this is a matrix that represents a vector, so

Â every vector can be represented in matrix form.

Â You break down vector components along three orthogonal axes, but

Â not every matrix represents a vector.

Â In this class, you will see sometimes things that we write this way without

Â this left superscript, and it's literally just a three by one stack of numbers.

Â We had three equations and it was easiest, again laziness,

Â it was easiest to write in a matrix form, because then we use classic matrix,

Â linear algebra math tools, and solve for this stuff, all right, so we see both.

Â