2D Cartesian VS Polar coordinate systems - what are they

Welcome back. In this lecture, we're going to look at different ways how we can determine an object's location, either on a 2D plane or in a 3D space. So let's start with a 2D plane. The system that we are most used to with is called Cartesian system. So this is called a Cartesian system, a rectangle system. So here this is our world. And what we do, we define our world, which is two dimensional right now. We define it as a set of many small rectangles or many small squares. So we chop our world into very small squares, and that's how we can measure the position of objects. So let's say if I have one object here and another object here, then of course I would have to use a vector to describe the position of the points. And for that, we use this Cartesian plane and we call it X and Y. These are the two dimensions, X dimension and Y dimensions, but we can really call them however we want. We can also call them u, h, then I don't know v s. So that's up to us how we call them. But the point is that we take our world and we chop it into little squares, and in order to measure position one, what we need to do, we need to go two units to the right. And then we need to go two units up and we would denote it like this p one equals and then the vector would be two and two, right? This is the x dimension and this is the y dimension. And the in order to find the position of P two, you need to go two units to the left in the negative x direction and then two units down in the negative y direction. So your P two would be. Minus two and minus two. But let me ask you a question. Is it the only way how we can chop our world? Can we chop the world in other ways? Well, there is a way we can also chop the world like this. So instead of saying that I'm going to go two metres to the right and then two metres up or two metres in the x direction and then two metres in the Y direction. I can also say, for example, that again, this is my world, right? So this centre here is equivalent to this centre here. So let's say I can say that instead of this, that, okay, I'm going to go these many meters here and these many meters here, I can say that. I'm going to go. Three meters to the right. And then 45 degrees counterclockwise or in the positive theta direction. So now I still have two dimensions, but my dimensions are different. Now my dimensions are radius. And angle. So if this is 45 degrees and this is three, I can say that I can go. Three metres to the right or in the other direction. And then I go pi over four radians. Counterclockwise. Four. Plus pi over four radians. And by the way, this is called polar coordinates. This point here in the middle, it's pole. And when your angle is zero, when your theta is zero, then this is your initial rate here. So this are here when your theta is zero. It's called initial ray. So I have drawn it here. So this is the pole, the middle point here. And the initial rate when theta equals zero radians. Now, I have two points here. P three and P four. And they are not the same, like P one and P two. There are different points. And the way you would denote p three and P four is like this P three would be a vector because you need two number lines to determine the position you would go, then three units along the initial array. So R is the initial rate when theta is zero and then plus pi over four radians. So three meters and plus pi over four gradients and notice the difference. Now in the x y plane, both dimensions were meters, but now our dimensions have different units. So we have the R dimension, which is meters and theta dimension which is either radians or degrees. I prefer radians. And now if you want to measure p four, then P four would be again you go three meters along the initial way, which is the way when theta equals zero. Three. And then either you go like this. So you go three pi over two radians or. You go three metres along the initial rate and minus pi over two radians. So you see if you turn clockwise then it's minus theta. And by the end of the day, it's just another grid, right? Just like in the x y plane, we had the grids and we chopped it into little squares. Well, here we have chopped our world differently, but still grid. Now our grid element is not a square. Right before it was a square. Now our grid element is something else. Are grid element is this and this. And the grid element itself is changing in dimensions. However, here we don't have length and width as our dimension. Here we have radius in meters and then an angle in radians. And so let me ask you a question. Are these two dimensions independent? Yes, they are, because I can go along the ah at any point in space without changing theta and I can also go along theta. I can turn around the center without changing or so I can go along one dimension without changing another, and therefore they are independent from each other. But tell me, why would we even bother using different kinds of grids? The answer is to make our lives easier. It all depends on your problem that you have to solve. For example, if you have a house and you need to measure the position of this point here, then it's probably easier if you just use the Cartesian coordinate system. Right. You know that. Okay. You can go a little bit to the right and then up and you will find the position there. For example, if you have a roller coaster, I can also put my RC to reference frame, for example, here and here, because here I need to measure the position to from this center of this circular roller coaster road. And then here I need to measure the position three from this center and then, okay, it's probably easier to work with the polar coordinate system here than with the rectangle coordinate system. The road is near circle. All right, so I put my center here and I go our meters along the initial way, and then I turn a certain angle. And the same thing here are meters along the initial way. And then I turn along to gradients to get to pi three. So that's why we use different coordinate systems.