2D Cartesian VS Polar coordinate systems - what are they

Mark Misin
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34:53:37 of on-demand video • Updated October 2020

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English [Auto] We'll come back in this lecture. We're going to look at different ways how we can determine an object's location either on to the plane or in the 3-D space. So let's start with a two deep plane the system that we're most used to with is called Cartesian system. This is called Cartesian system a rectangle of system. So here what we do this is our world. Right. 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 various 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 then mention and Y dimensions but we can't really call them however we want. We can also call them you h. Then I don't know the answer. So that's up to us how we call them. But the point is that we take our word and we jump into little squares and in order to measure position 1 What we need to do we need to go to units to the right and then we need to go to units up right to use to the right and 2 units up and we would denoted like this one equals and then the vector would be 2 and 2. Right. This is the I mentioned is the y dimension. And then in order to find the position of P2 you need to go to units to the left in the negative x direction and then two units down in the negative wind direction. So your P2 would be minus two and minus two. But let me ask a question. Is it the only way how we can chop our world into small little squares. Can we chop the world into. In other ways. Well there is a way we can also chop the world like this. So instead of saying that OK 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 wind direction I can also say for example that and again this is my world right. So this center here is equivalent to the center here. So let's say I can say that instead of this that OK 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 on the positive theder 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 I can say that and this is three I can say that I can go three meters to the right or in the art direction. And then I go over for radiance counterclockwise or plus PI over four radians. And by the way this is called Polar coordinates. These are called Polar coordinates. And so this point here in the middle it's pole. And when your angle is zero when your seat is zero then this is your initial rate here. So this are here when your seat is zero. It's called initial array. So I have drawn it here. So this is the pole the middle point here and the initial rate when feet equals zero radians. Now I have two points here. P3 and P4 and they are not the same like piu one and P2. There are different points. And the way you would denote P3 and P4 is like this P-3 would be a vector because you need to number lines to determine its position. It would be a vector and you would go then three units along the initial array. So R is the initial array when seat 0 3 meters long been a. And then plus Piver for radians. So three meters and plus pile were four radians and not is the difference. Now in the x y plane both dimensions where meters. But now our dimensions have different units so we have the R dimension which is meters and saeter dimension which is either radians or degrees for radians. And now if you want to measure P4 then P4 would be. Again you go 3 3 meters along the initial ray which is the re 23 0 3 and then either you go like this. So you go 3 PI over to radians or you go three meters along the initial rate and minus PI over to radians. So you see if you turn clockwise then its minus minus theater. And by the end of the day it's just it's just another grid. Right. Just like the in the x y plane we had the grids and we jumped into little squares. Well here we have chopped our world differently but still a grid. Now now our grid element is not a square right before it was a square. Now our grid elements something else on our grid and does this and this and OK the limit itself is changing in dimensions. However Well here we don't have the length and width as our dimension here we have a radius in metres and then an angle in radians. And so let me ask you a question Are are these two dimensions independent. Yes they are because I can go along along the or at any point in space without changing Seeta and I can also go along sæter. I can turn it 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 different kinds of coordinates systems or grid systems. Well just the answer is to make our lives easier. It all depends on when a problem that you have to solve. For example if you have a house and you have you need to measure the position of this point here then it's probably easier if you just use the Cartesian coordinates system right. Then you know that OK you can go a little bit to the right and and then up and you will find the position there. For example if you have a roller coaster ride and you can These are called axes and you can place the original of it wherever you want. In the world. So I can place one reference frame. They're also called reference frames. So I can put my x y reference frame here but I can also put my R C reference frame for example here and here because here I'll need to measure that position to right. And here any too. But from the center of this circular road across the road. And then here you need to measure the position three from the center. So I put one reference frame here and then ok it's probably easier to work with with the polar coordinates system here then with with a rectangle a coordinate system. So because it's the road is nearly circular. Right. So I put my put my center here and I go our meters along the initial array and then I turn a certain angle and same thing. Here are meters along the initial array and then I turn and then I turn along Seatle radians to get to PI 3. So that's why we use different coordinate systems. But let's go to do three dimensions now.