 In this video, we're going to talk about Cartesian coordinate systems. This is a set of coordinate systems that was proposed originally by Rene Descartes and has a particular set of rules for helping us determine locations in space. Now, as we've discussed in a previous video, one of the things we want to do to describe locations in space is to define our basis. And there's three parts to our basis, determining an origin or starting point, determining an orientation, and determining a scale. So we're going to walk through three different levels of the Cartesian coordinate systems and we're going to start in one dimension. In a single dimension, we're basically interested in something that only has one direction associated. It could be its position of places along the river or maybe lengths of string or wire, something along those lines. And so I'm going to go ahead and draw my one-dimensional system as a straight line. Now, following our definition of basis, the first thing we want to do on that line is describe an origin. So I'll pick a spot on that line, a reference, starting point. Great. Now I've defined an origin and everything that we describe on that line is now going to be denoted in reference to that origin. Second thing we need to describe is an orientation. Well, it seems like we might already have an orientation because we drew the line, but there is still a piece that we need to do. What we're going to do is decide which way is positive and which way is negative. This way has a certain, well, preference to it because we're going to use different numbers and our number systems, as you learned way back in elementary school, have a direction that's associated with them as well sometimes. So in this case, I'm going to go ahead and draw a little arrow here and put a plus next to it indicating that that's going to be my positive direction and that positive numbers might be associated with that. And over here, I will put a minus in the other direction. And so our orientation, our positive orientation is essentially starting at the origin and going to the right. Last but not least, I need to go ahead and define a scale. Often we'll associate the number zero with the origin, handy because there's an O for zero and also for the origin itself. And then I'm going to make another mark somewhere along here that's going to represent a single unit. That one unit that gives me a scale and often I will associate that line with the number one. Now we all know the amount of distance between zero and one is one and so one unit is going to represent that little distance in there as well. And that creates our scale. Well now, what this allows me to do is take any place on this number line in this system and associate a place with a number. For example, if I put a place right there, hmm, how would I describe that place? Well, assuming I know my basis, which I've drawn, I can now calculate a number that's associated with that place. And maybe I look and I say, okay, the units are about maybe about that big as two units and maybe there's about three units. And so that might be something like 3.5 units, whatever our unit is, is represented by that location. So every location corresponds with a number. And one number describes a place in space in this one dimensional basis. Notice with this, I can describe all the real numbers. Every number that you can think of that's known as a real number. Yes, there are imaginary and complex numbers. We're not considering those, but all my real numbers can be represented by this line as long as I make it long enough. Or I can get in really close and zoom in and find a number that's detailed. 3.5, 7, total 9, 8, 17, whatever put in numbers and they go on forever can be represented by one real. And this is what makes us a one dimensional space. One number, one place in space. Now let's consider a two dimensional system. In 2D, once again, we're going to think about what is encompassed in two dimensions and usually what that means is something like, oh I know, the glass that I'm writing on here or a piece of paper. Something that's flat where we care about two different directions but we don't care necessarily about the third direction very much. So in my two dimensional system, I'm once again going to build my basis. I'm going to start with an origin. Here I'll place a starting point in space, label it with that O which is both origin and zero. We often use zero as our starting point although we don't have to. You can use different numbers as a starting point. It just tends to confuse things a little bit. And now I need to establish my orientation. However, my orientation is a little more complex here and I'm going to establish my orientation by drawing an arrow off to the right. This is very typical in math classes to start with off to the right and we'll label that maybe our x-axis. You've probably seen something like that before. But I'm going to go ahead and extend that a little bit further and make a number line just like my one dimensional system with positive to the right and negative to the left. But that's only one dimension. It looks just like my one dimensional system. Now what I'm going to do is add a second dimension. And this is where René Descartes and the Cartesian system have particular specifications. What they want us to do to find a system is to make sure we make our second set of directions perpendicular at a right angle to the first set. So now I'm going to define up as positive that's very typical and down as negative in my two dimensional system. So now I have my origin at the middle of these two axes. Again it doesn't have to be there but it's simple as if it is. I have an orientation described where I know that I'm looking to this I want this to be up and this to be to the right. And then the last thing that I need to do is establish a scale. As it turns out we could establish a different scale for each of our directions if we want to. That tends to make things a little difficult but it is doable. In this case I won't choose to do that. I will go ahead and draw a unit here and we'll put that as a one unit in that direction and we'll simply do something that's approximately the same. It's not going to be perfect here but also a one unit there. Now I have a scale in both of my dimensions and the scale happens to be the same which makes my math again easier. But it doesn't have to be the same. Now I can represent a point in two dimensional space I'll go ahead and draw one. Let's pick a place out here in two dimensional space. Here is a point in my two dimensional space. Well I need to describe where that point is that's the entire purpose of this coordinate system and the bases and I'm going to describe that by first considering this horizontal axis how many steps over do I need to go there? So let's see here it looks like I need to hop over one, maybe two, maybe three and we're going to say that's just over three maybe it's about 3.1. So we'll associate a number with the horizontal direction of 3.1 units. However that's not enough to define that place in space because there's a whole bunch of things that can also have that same horizontal distance. So now I'm going to do something that you probably have probably seen before now I will also move in another dimension I will go up one and let's see here about where this is it looks like it's a little bit less than two so we'll call that maybe 2.8 2.8 units and notice I'm changing the color there to distinguish between the two. So now I have two numbers two dimensions making this two dimensional that describe that point in space so to describe a point in space in the Cartesian coordinate system I'm going to need two values that are associated with lengths. Notice that makes this a R2 R being real numbers the first number is all the real numbers represented on the horizontal and all the real numbers represented on the vertical and we'll write that as R2 space that's a two dimensional space the other thing that's important here is that there's a difference between 3.1, 2.8 and reversing them being 2.8 and 3.1 which would be something that would probably be up in that space in that area so in other words it's important to keep the order of these two numbers and you will probably recall hearing the term ordered pair that the numbers together will be kept in order and that order is determined by our description of our basis. Now let's discuss three dimensional space now three dimensional space poses a little bit of a challenge for me drawing on a two dimensional board so let's consider how we're going to represent three dimensional space which includes this idea of front to back while I only have a two dimensional space to draw on but we've been doing this for a while so let me start with another two dimensional space this piece of paper and as you can see this piece of paper probably lines up pretty well with our original two dimensional space well now I'm going to take that paper and think about what it looks like if I lay it and change it and move it into my three dimensional space and if you look at it you can see that my little arrows that are drawn orient themselves differently depending on how I'm orienting this page in space I'm going to assume that we're looking at this piece of paper at a little bit of an angle where you can see it and you can see those arrows and I'm going to try to draw them as you see them here on my screen again I'm going to create my basis by creating an origin here is my origin my starting point and I will create my first dimension here is my first dimension and we'll draw a positive to the right and draw a negative to the left that's establishing my orientation in one dimension then I'm going to establish my orientation in a second dimension and the way we're going to do that is I'm going to draw this off at a bit of an angle now on my two dimensional space this just looks like a line at an angle but if you can kind of picture and you've probably seen this because we do three dimensional drawing all the time think about this thing going off into the distance this is going coming back into the page coming back toward me we'll call that a way a positive direction and this over here will be my negative direction and you can sort of consider that I don't know if I can get it quite right but if I take my pages here I can make my angles almost look like that where one's going across and one's going at a bit of an angle do notice that this is even though it doesn't look like it on my two dimensional drawing right angle to fit our Cartesian coordinate system which assumes that all these things are at right angles now my third dimension I'm going to describe as being up and down and we'll use the convention of up being positive and down being negative so now we have three dimensions in space two of them are represented just like 2D right to left and up to down and this other dimension that represents back to front and hopefully you can see that this coming out here represents coming back toward me and that there represents coming in toward you so we've established our origin we've established our orientation again I will note right angles are important here so there's right angles between each of these sets maybe this is also a right angle here right angles associated with all of these pieces and now I need to establish a scale once again the scale does not need to be the same in every dimension however I will do my best to go ahead and keep it pretty close to the same as that's typically what we will do when we're describing things in space let me make my unit I'm going to make this unit just a little bit longer here is my one unit and that established my scale I will also draw that unit on my vertical axis and then also on my axis that's going in and out of the page now notice how you draw that one going in and out of that page there are some processes that you may see later as we learn how to do engineering drawing with how long you make that to look like it's the same distance I'm going to try to approximate it to be about the same length as the others but how you choose to draw that length is determined by some certain processes in drawing alright so now we have each of those units in our dimensions and we've established our basis well now that we have a three dimensional basis we can now describe things in three dimensional space where is something located now again this is going to be a little confusing I'm drawing it on two dimensional space but it would be nice if I could put it out here somewhere or in toward you somewhere I can sort of do that with where I am but let me go ahead and locate a point out here in space so there is a new point in space and I'm going to figure out where that point is in three dimensions this is again a little bit harder because of the two dimensional nature so I'm going to do my best to sort of visualize where this this is I'm going to assume that it's over a certain amount we're going to say that it's comes over one two well let's see here we'll say that's about 2.1 units and then we're going out a certain amount until we're directly below where it is and actually I want to make that a slightly different color so we'll go along that line it looks like it's about one maybe one and a half units and then we'll go up a certain amount and that appears to be probably about half a unit or so notice I could of that point could also be a point that was over here back this much and up a very different amount that's the problem with two dimensions and three dimensions is that this same point drawn on the board could represent a whole bunch of different ones but in this case I'm going to say this particular point was over about 2.1 units to the right from back to front it was away and into the page about one point we'll say 1.4 units and then we went vertically about half a unit 0.5 units notice the order of these three numbers is important 2.1, 1.4 or 0.5 is very different than 0.5, 1.4 2.1 or any other permutation of those things so their order is important again we now have an ordered group maybe not an ordered pair but an ordered group of three real numbers and so now we're in three dimensional real space we describe Cartesian coordinate systems we describe our basis again origin orientation and scale the main features of the Cartesian coordinate system is that our orientations consist of three number lines all of them perpendicular to each other terminology for that is something called orthonormal and then the three numbers the three dimensions we use are each associated with one of those number lines and they are all unit numbers they're all numbers that have a unit associated with them are associated with the scale of a length so we have one length, two length three lengths along systems that are all orthonormal and that creates for us a Cartesian coordinate system