 So the topic of coordinates is a solution to a somewhat complex problem. Say I have a point someplace in space, and what I'd like to do is I'd like to be able to specify where it is. Now, I could say, it's right here, and maybe if you don't get that it's right here, I could shout louder, or I could do, I could wave this over the point to say, look, look, look, here it is, here it is, here it is. But at some point we want a somewhat more sophisticated way of doing that, because that will only work if you can actually see what I'm seeing, which generally requires you to be sitting exactly where I am and looking at exactly the same thing I'm looking at. So let's see if we can define a coordinate system to be able to specify where that point is located. Now, in order to do that I need two things. I need a reference point. I need the place where the coordinate system is going to originate. Now, in the personal universe we have a very egocentric coordinate system where the origin is me. I'm right at the center of the universe. However, if we don't want to do something like that, because of course it gets complicated when you have more than two people in the universe, then we agree that some location is going to be our origin. So maybe I'll put the origin right there. So there's the origin of our coordinate system. And then the other thing we need to specify is some sort of reference direction. So for example, I might specify that this line is going to be my reference direction. So I have a specific point, the origin of the coordinate system, and then I have this reference direction that I'm going to use to describe where this point is located. And one thing that you might think about here is that you want to give directions for how to get from the origin of the system to the point of interest. So how do I give those directions? Well, for almost anybody on the planet, the most natural way of giving directions for how to get from the origin to a point is going to look something like this. So we'll make use of that reference direction. So we might say something like, okay, so start off by facing this way, face this way, and there's your reference direction being used. And then turn until you face the point. So I'm going to turn, I'm going to do a little rotation, and now I'm looking straight at the point. And it's worth remembering that what I've done is I've specified, make a specific rotation, turn a specific amount, and we'll have to measure that amount somehow. But the idea is that it's some specific amount that we'll be able to determine. And again, if you imagine that you're giving somebody directions, now that they've turned, they're now facing the object of interest. And then the next thing I might do is I might say, well, just walk that way until you run into the object. Or if I want to be a little bit more specific, I'm going to tell them how far they need to walk. So they walk out along that direction and they get right to the object. Now those two numbers are all that we need to locate any point. And so we call them the coordinates of the point. And we express them as an ordered pair. For a variety of reasons, we give the distance first, and then we give the angle as the second term of the ordered pair. And here's the important thing to note. If I get either of those wrong, I go to the wrong place. So for example, let's say that I start with my reference direction and I turned over to there. And I walk the right distance, but I'm not at the correct point. Or perhaps maybe I get the angle right. So I'm going to turn until I'm now facing the point. But then I don't walk the right distance. I walk out to there and I'm here and I'm saying, where is the red point? And the answer is I haven't gone far enough. I need to go all the way out. And to distinguish this way of specifying where a point is located. So again, what we've done is we've turned and walked. So this is a turn and walk specification of the location of a point. We say that this type of specification gives us the coordinates of the point in polar coordinates. Well, let's try an example of it. So I'm thinking of a point. And it's going to be located. My angle of rotation is going to be a quarter turn counterclockwise. And make that quarter turn counterclockwise and then walk out two length units from that place. Now, in order to do this, we need our reference point. So there's my reference point. And I also need my reference direction. So I'll draw my reference direction line. And by convention, we always start pointing to the right. We always start facing to the right. And that is our starting point. And so when I take that quarter turn counterclockwise, I'm going to turn counterclockwise from that initial direction. So I'll make a quarter turn clockwise. And then I'm going to walk in that direction two units outward. Now, in order to really do this correctly, we have to know how big the unit is. But let's say it's about that far as one unit. So I'm going to walk in this direction two units outward, get to that spot. And that must be the point we're talking about. So there's my point that is a quarter turn counter clockwise and two units out. Well, what happens if we're trying to locate a point? But for whatever reason, we're not able to walk straight there. So for example, if you're walking in a city, Manhattan, trying to go from your reference point straight to where you want to go. If I have my reference point here, if I want to go straight to the reference point, maybe I have to cut through some buildings, maybe some subway tunnels, drive over some streets. It might not be physically possible to go from here to there. So what can we do? Well, we still need that origin. We still need our point of origin. And we still need a reference direction because we can't describe how to get there unless we have both the origin and the reference direction. So this time, we're going to walk some specified distance along our horizontal until we get to a point that's directly underneath where we want to be. And then we're going to walk vertically until we arrive at our target point. So this time, we have two coordinates that specify the location of our point. That would be the horizontal distance that we travel, however far that is. And then the other coordinate is going to be the vertical distance that we travel. And again, we'll specify both the horizontal and the vertical distances as an ordered pair. Horizontal first, vertical next. And again, these are also our coordinates. So we say that what we have is specification of the point in rectangular coordinates. Or we might also call these Cartesian coordinates because they were invented by René Descartes among other people. So for example, say I want to locate the point 2, 3. So again, reading that, first off, a couple of things that are worth noting. The enclosing symbols for our coordinates are parentheses, which indicates that we're looking at rectangular coordinates. And that tells us that 2 is the horizontal distance, 3 is the vertical distance. So again, to make use of these coordinates, I need to put down the origin. So how about right there? And then I need to put down my reference direction. And so my horizontal coordinate 2. So that says I should go 2 horizontally. So let's take a walk. And then my vertical coordinate 3. So again, we'll take a walk. We're going to go out 2. And then we're going to go up 3 units from that point. So I'll take a walk. And I've now arrived at the point where I want to be. And so I'll go ahead and mark it. So again, another example, let's try and locate where the point 3 negative 1 is. So again, I want to set down my origin and my reference direction. And because my horizontal coordinate is 3, I want to walk out 3 units. So let's go ahead and take a walk from there. And this negative 1, it seems reasonable to interpret that as going 1 unit downward. So I'll take a downward step, 1 unit. And what's that done? Well, let's put me 3 units over, 1 unit down. And those are the coordinates of the points. So the point I'm interested in is right where.