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Coordinate Systems - Overview

In mathematics, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the "x"-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The most common coordinate system in modern mathematics is the Cartesian coordinate system.


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Video Transcript

welcome to our video on reviewing coordinate systems. This is part of unit zero and our math review in this section. We're going to talk conceptually about the different coordinate systems that we use in physics because there is more beyond just the standard Cartesian coordinates. But let's go ahead and start talking about that now when I say Cartesian coordinates, what I'm talking about here are is what I have on the screen the standard X Y coordinate system and in Cartesian coordinates. As you're probably familiar, what we have is each of these good lines corresponds to a number. So this is the one the two, the three, the four and the extraction. Likewise, in the while will have the 1234 picking this spot in the middle to be the origin 12340.0 Now, when we write points in Cartesian coordinates, we usually write them like this with the X, the independent variable, and then the why the independent, The dependent variable s O. For example, if we were going to look at the 00.46 we'd count toe 1234 and then we'd count up 6123456 and we'd make our point there. Um, now, Cartesian coordinates are extremely useful in physics. For one, they're great for drawing vectors. For example, if I were to draw a vector, I'd go here. And the reason they're great for drawing vectors because not only can I draw the vector itself, I can also draw what are called the components of the vector really easily. That's the point. The pieces. If I were to draw too little vectors that we're going to add up to that final vector, I could draw them like this, and you can see that one of the components is all in the X direction, and one of the components is all in the Y direction. And we love this is Physicist. Um but it's This isn't always necessarily the most useful coordinate system for solving problems as such. Sometimes we want to check out one of our other options. So we're gonna look at some of those other options today. Eso First of all, we're gonna look at what is called polar coordinates. So polar coordinates, uh, kind of can be visualized, like Cartesian coordinates. You see, I've highlighted the horizontal and vertical axes here, but notice instead of the standard X and y grid lines. What I have here are a series of concentric circles because in polar coordinates, what we're actually interested in isn't X and y, but what we're going to call our and Fada, that is to say, the radius or the distance from the center at which our point is and the angle. So, for example, if I were to choose this point right here, okay, then I'm I would have a radius of one, 2345 So my radius would be five, and then my angle, I happen to know this is 15 degrees, so this would be the position of this point on the polar coordinate system. Now you can see that this is this is pretty different from our Cartesian coordinates, but it's also related. Uh, if I were to think about this in terms of X and Y, in fact, I could just draw a triangle here I could say, Oh, I have a triangle that looks like this as a hypotenuse of five and an angle of 15 degrees, and we've already viewed well, we know enough trigonometry to be able to solve for this, we're going to find that this is gonna be five times cosine of 15 degrees, the X coordinate and the Y component is going to be five times sign of 15 degrees. And in fact, this shows us the relationship between the Cartesian coordinate system and the polar coordinates system. Um, we know that X is equal to our times, Kassian Fada and why is equal to our times sign if they did now. Importantly, this is assuming that we are always measuring data is being measured from the X axis, the positive X axis. So we know this is Arteta. Even if we went all the way around into some funky quadrant, like all the way over here. Okay, we would have to measure that entire angle. We wouldn't we? We could measure the negative, but we just have to make sure we call it negative. But a positive angle means we're going around counterclockwise all the way around here. Um but so these air, this is how we can get our Cartesian coordinates out of our polar coordinates. Likewise, If I were to go back and look at this thing, I could say, Oh, well are, then is going to be equal to the square root of X squared plus y squared. This just comes from the Pythagorean theorem and theta. Okay, so let's say they're going to be well, we could use tangent of data, which would be opposite over adjacent, which in this case, is going to be arc tangent of opposite over Jason. That's gonna be why Over X. So now we have two ways to convert here. We have one way to convert from, uh, the polar coordinates to the Cartesian coordinates. And then we have another way to go from the Cartesian coordinates into the polar coordinates. You have to be a little careful when calculating data. Make sure you know which quadrant you're in and always make sure that your data that you calculate is inside that quadrant. You just have to pay attention a little. I really invite you to try a couple of try a couple examples on your own, and we'll do some later here. Okay, so this is one other alternative coordinate system that we can look at the polar coordinates. Ah, third one that occasionally comes up. They're not so much an introductory physics. But a lot later on in more advanced physics is what we call cylindrical polar coordinates. So this is cylindrical polar coordinates. So notice what I've done here is I kind of took the polar coordinates, and I flipped him on their side and stacked them vertically. Okay, so we still have our in Thatta as variables, But then we have a third variable to which we're going to call Z. Okay, so we have our theta and Z and Z is the number here. So if this is the zero Z, this would be one Z and two Z nearly have negative one z and negative to Z. So Z is telling us how high we are in a vertical direction. Okay? It's telling us which of these planes were in. Meanwhile, R and theta are still gonna pick exactly where we're going to be Now. It's hard to draw the axes here and get it to look right. So I have ignored it. But imagine if we had a horizontal axis, okay, that dictates where the zero degree line is okay, and then we can rotate around it. Then these concentric circles again are illustrating our our so we'd have a point that would look like this. Okay. And if we were going to draw the total vector coming from the 000 point, the origin here in the middle, then I would have to draw all the way up there. Okay, So are our in this case, then would be this distance here, whatever that I think we're in. 123 We're in the third concentric ring. Okay, so we have on our equal to three there, and then we have a theater. We'll just say Arteta's 15 degrees. It's a little hard to see at this angle, and then our Z would be to Okay, so this is pretty easy toe handle. Conceptually in terms of just all you're doing is returning the polar coordinates on their sides and then stacking them in a grid pattern. Um, now, don't be intimidated. Just because we've added in another dimension here, it's actually pretty easy toe handle. A third dimension in physics and in math, it's just a matter of tracking all of your variables. A little more difficult to visualize is our last coordinate system, which is called are Spherical polar Coordinates. Now it's very difficult to draw on a two dimensional surface. But I'll do my best year, so we have spherical polar coordinates. So our variables and spherical polar coordinates are gonna be our fada and five So fi here is going to dictate another angle here. What I've done is I've got three coordinate axes. So I've got my ex and my why we could think of here. And then I'm gonna have, um I'm gonna have my Zia's. Well, okay, so on X, y and Z So if I were to correspond this to Cartesian coordinates, it would look like this. Okay, so this is a Cartesian coordinates In order to measure polar coordinates, though, what I need to do is I need to find are so that's the distance from the center. So if I were to pick a random point here it be the distance there would be our and then if I were to project that down, or rather if I were to look at that in the X Y plane. Okay, so the angle coming off of X, that would be my Thatta. And then what fi is fi is actually a new angle measured from the Z axis. So in this case, if that points in the X Y, plain thin Z would be zero and five would be 90 degrees. If it were out here, then I would measure fi from the Z axis. We'll talk about how to do that in the later video.