 Hi, I'm Zor. Welcome to UNISOR Education. Continue talking about solid geometry. I would like to discuss today the coordinates in three-dimensional space. Well, this lecture, as you know, is part of the educational website UNISOR.com. That's where I suggest you to watch this lecture from, because there are very important notes. Also, the website allows you to basically enroll into a specific course or part of the course or a specific topic and take exams, which is very good for, like, home education or so-called flip classroom. Anyway, so coordinates in three-dimensional space. Quite frankly, I'm sure that many of you actually know what I'm talking about. Well, if I will draw a picture, something like this. And so this is x, this is y, and this is z. You know what this coordinate system actually is. Nevertheless, I would like to talk a little bit about this, maybe on a little bit more rigorous level. And probably I will also introduce, maybe in another lecture, other systems, not Cartesian systems like cylindrical coordinates and spherical coordinates, but that's subject of other lectures. So today I will be talking about coordinates and specifically about Cartesian coordinates. Well, first of all, why do we need coordinates? Well, you might say that we need coordinates just to be able to identify our position in the world relative to some system, some reference system. Well, which is true. Also, we need coordinates to be more numerical about our geometrical properties. For instance, we are talking about, let's say, a sphere as a set of points which are equidistant from a center. Now, this is purely geometrical explanation. Now, is there any algebraic expression of exactly the same property? Yes, it is. It exists. And it allows to do something relatively sophisticated proofs of certain properties. So there are certain things which cannot be actually addressed purely geometrically and we resort to coordinates and numerical representation of certain geometrical objects or their properties in order to be able to better investigate basically what these properties are or what these geometrical objects really are. Especially, it's important in those cases when geometry is not applicable at all and I'm talking about higher-dimensional space than 3, 4-dimensional, 5-dimensional, whatever, 39-dimensional space. Now, these do exist. It's not in our traditional geometrical imagination but they exist in certain theories which require this type of approach. And there are no geometry which could support actually certain properties or certain statements about geometrical objects. So you have to resort to some numerical analysis and that's where coordinate system, especially Cartesian coordinate system, is very useful. So back to our Cartesian system. So what's the definition? Now, there are many ways you can define Cartesian system of coordinates in 3-dimensional space. Here is one of the ways you can actually do it. First of all, you definitely need a fixed point which you can call origin of the coordinate system. Now, once this point is chosen, then, as I was saying, there are three coordinates which we have to define somehow. So first of all, we have to choose the axis. Now, I would prefer to start from the axis which differentiates 3-dimensional from 2-dimensional case which is traditionally called z-axis and traditionally it's drawn vertically on the board or on the paper. So this is the axis which I am talking about. Now, axis must be directional. So we have to choose a specific direction. Now, next what we do is we choose a plane which is going through this point of origin and it's perpendicular to our axis z. Alright, so let's draw it something like this. Now, in this plane, we have to choose two mutually perpendicular axes which I will call x and y. So let's say this is my x-axis and this is my y-axis. Well, actually this is a solid line because it's on the plane. So now, I have three axes which are mutually perpendicular to each other. Now, z is perpendicular to both x and y because z is perpendicular to the plane and I will call this plane xy-plane. Now, x is perpendicular to y because that's how we have constructed and y is perpendicular to x and both of them are perpendicular to z because they belong to the plane which is perpendicular to z. So all of them are mutually perpendicular, all three axes. Now, what also is important is direction. Traditionally, the direction of these axes are chosen in such a way that if you will look from the positive direction of z towards the x-y-z plane, then the direction from positive direction of x to positive direction of y would be counterclockwise. This is a traditional way of representing graphically these three axes. So all three of them are mutually perpendicular and the directionality is chosen in such a way that from the positive z if you look down from x to y direction from positive x to positive y direction would be counterclockwise. So we have defined three axes. Now, what's next? Next we have to define a unit of measurement which we take as one. Whatever that segment actually is. Now, obviously that's sufficient to define three numerical characteristics of any point. Let's say we have a point P and I would like to define a position of the point P using my three axes and certain real numbers. So the way to do it, again, there are many ways to do it, but here is the way which I have suggested, which I'm suggesting to you right now. You draw a perpendicular plane from this point P to each of those three axes. So perpendicular to z would be something like this and it intersects at point, let's say, A z. Now perpendicular to y, that would be this, plane can look something like this, perpendicular to y and that would be the intersection with y. And finally perpendicular to x, something like this and the plane maybe like this, whatever. And that's perpendicular to y and A x. So I have three points from the point P I have a perpendicular plane to z, to y and to x and they all intersect the corresponding axes at certain points. So now, if this is my origin this point all, I have three segments. Now each of those segments has certain lengths using this unit of measurement. And also I will assign a sign if my point A y or A x or A z are on the positive branch of the axis. It will be a plus sign. If it would be on a negative, for instance, something like this, if I will have a point A here, then its coordinates can be negative z, maybe positive x and negative y, something like this. So these three will go to my x, y and z coordinates which are the lengths of these segments with a sign depending on where my point of projection actually is. Now is this the only way? Of course not. I can do it differently and get exactly the same result. Here is another variation. I will drop a perpendicular from point P. Actually, let me draw another better picture. Now it will be exactly the same result. Let's say this. So that will be my x, that will be my y and this will be my z and the point P is somewhere here. So I drop a perpendicular to a plane, let's say this is a plane, x, y plane. So I drop a perpendicular to the plane and from this point, I will drop a perpendicular to the x and perpendicular to y. So this is A, this is A x, this is A y and I also have to drop a perpendicular to z. It will be A z. So that's another way of doing this. But to tell the truth, it's exactly the same thing. As an exercise, you can actually prove that whether I'm doing it this way, dropping perpendicular to x, y plane and then from this point, perpendicular to x. Or I will have a perpendicular plane which goes perpendicularly to z, to y and to x. I will always get exactly the same points, A x, A A y and A z. So these points are actually projections of my point A onto each of the axes. In any case, we come up with three numbers which are, as I was saying, the lengths of these three segments with a proper sign depending on the position of these projection points. Now, it might actually look like I have chosen the axis, the z axis as some special axis. Actually, no, they're all absolutely interchangeable. I can start with x or y. It doesn't really matter. What does matter is that in our three-dimensional world we need three axes and three numbers in the Cartesian system. That's what actually gives the name three-dimensional world, because there are three characteristics of every point. If you have some kind of a frame of reference, you need three numbers to identify the position of this point. In other non-Cartesian systems like cylindrical or spherical system, which I will be discussing in the next lectures, you will need different numbers. They're probably angles or something, but they're still three. That's what's quite interesting actually. No matter how you approach this task of identifying the position of a point, in three-dimensional world you need three numbers. Many different ways to do it, but you still need three numbers. Dimensionality is really a very, very important property. I would like to mention one very important thing. With each point I was just explaining how I can connect this each point to a triplet of numbers. Now, my next question is, is it one-to-one correspondence between all the points in three-dimensional space and all the triplets of real numbers? And the answer is yes. So for each point I will get some three numbers, and for each triplet of numbers I can get the point. Now, let me just go in reverse. So if I have three numbers, I'm basically fixing the position of three points on three axes. The point actually is on the length equal to the absolute value of corresponding number and the position on one side and the positive side of the axis or a negative based on the sign. So I have three points. Now, if I have three points, three projections, does it define uniquely my position A? Yes, absolutely. And again, to construct it, you might do something like this. Within the X, Y area, using AX and AY, you draw a couple of perpendicular and you will hit one particular point. That's a projection of my point A on the XY plane. Then you have a perpendicular to this. And again, depending on the sign, you go up or down on the length, which is actually specified by absolute value of Z. So you can always construct. From a point, you can construct a triplet. From a triplet, you can construct a point. Another story is, and that's much more subtle and I would say delicate issue in mathematics, are all the points in space can be converted into triplets of number or are all the triplets of numbers can be converted into a particular point. So that actually, as I was saying, a delicate object intuitively is obvious. Now, if you want to do it like absolutely rigorously, well, that's not so easy. I mean, there are certain axioms you have to use and I would rather stay away from this issue and let's just concentrate on intuitively obvious fact that any number can be converted into a segment, basically, with the length of which is specified by this number. And then, from the end of this segment, we can actually construct the point. So that's quite an obvious statement and that's what I would like to stop, actually, not go any further as far as the rigorousness is concerned. All right, what have I not covered? Okay, the names. All right. The X coordinate is called abscissa. The Y coordinate is called ordinate and the Z coordinate is called applicable. So this is X, this is Y, and this is Z. Well, it's just the names. I mean, so if somebody says, okay, abscissa and such-and-such, ordinate, such-and-such, and applicable, such-and-such, it means X, Y, and Z coordinates in the curfusion system. All right, what else is interesting? Okay, now there are a couple of exercises which I actually would like to do with you just to have a feel of why exactly these coordinate systems might be important and how they can be used. Okay, so I will just ask a couple of questions and I will answer it myself. All right, so let's take the origin of coordinates. Now, this is a point and as any point it's supposed to have some coordinates. So what are these coordinates? Just think about it. If you will draw a perpendicular to every axis, the plane perpendicular to every axis it will go through the same point. So basically it's 0, 0, 0, right? So if I will draw only the positive directions to make my picture a little bit cleaner. So this is my original coordinates. Plane perpendicular to Z would go through this. Plane perpendicular to Y goes through this and plane perpendicular to O goes again through this point. So three planes are intersecting in the same plane. So the length of each segment on each axis is 0. Okay, next question. Can I numerically or algebraically explain that I would like to talk about a geometrical object which is XY plane? So what are the characteristics of three coordinates of the points which belong to XY plane? So every point here. So if I would like to say, okay, what is the equation which kind of connects XY and Z together? Which describes the whole plane with the whole plane which is XY plane. Well, let's just think about it. X can be anything because the point can be anywhere so projection of the point to the x-axis can be any. Positive, negative, doesn't matter. Y, same thing. And only projection on the Z, what is the plane which is going from this point perpendicular to Z? Well, the same XY plane, right? Because X is perpendicular to Z and Z perpendicular and Y perpendicular to Z. So the whole plane XY is perpendicular to Z. And this point belongs to this plane. So if I will have a plane which is perpendicular to Z, now this plane is XY plane. So any point on the XY plane, if I will draw a plane perpendicular to Z, it will coincide with XY plane and it will intersect Z exactly at point zero. Which means the Z coordinates should be equal to zero while X and Y can be any. So this is an equation which basically represents all the points which are lying on the plane XY. I hope I explained it relatively well. So again, how do I determine coordinates? I'm taking this point and draw a perpendicular plane to all three axes, right? And wherever these perpendicular planes intersect, that would be my projection. So projection from this point can be somewhere on X plane, somewhere on X axis, somewhere on Y axis. But on Z axis it will be always zero because the point belongs to this plane which is already perpendicular to Z. So this is equation of the XY plane. So the plane has equation. Incidentally, any equation of this type is a plane which will go through the origin at some angle probably since all my... Now I'm not going to prove it right now, but basically that's what it is. And if you will have some kind of a constant here, that might be actually any plane in space, in three-dimensional space. Alright, so next exercise. Angle bisector between Y and Z. Okay, so within the Y-Z plane I have an angle bisector and I would like to find an equation, an algebraic equation of X, Y and Z which represents all points on this. Now let's just think about it. Since this particular point anywhere on this bisector belongs to Y-Z plane which is perpendicular to X axis, it means that projection from any point on this bisector within Y-Z plane, projection onto the X axis would always hit point zero, right? Because the entire Y-Z plane is perpendicular to the X and intersect at origin of the coordinate. So the plane which is perpendicular to X axis from this point is actually Y-Z plane which intersects at zero, which means X is equal to zero. That's my first equation which basically narrows down all the points in the universe to only the points which are lying on this plane. But now I have an additional requirement. The requirement is that I am on the bisector of this angle, right? Now bisector obviously means that these two coordinates, Y-coordinate and Z-coordinate are the same, right? Since these angles are the same and this is common hypotenuse, it means that our cacti are congruent to each other. So the second equation which is also supposed to be part of the algebraic expression of the property that the point lies on the bisector is Y is equal to Z. So together as a system, these two equations define this line, the bisector of this angle between Y-axis and Z-axis. All right. Now the third example, fourth example, whatever, I would like to present is, I would like to describe somehow a sphere, not just any sphere, but a sphere which has a center exactly at the origin of coordinates and the radius R. Well, okay. Let's just think about how can I express this. Now, we know that the sphere is a set of all points in three-dimensional space which are equidistant from its center. Now, center is at the origin, right? So let's just take some point, let's say here on a sphere, on my side of the sphere. Now, its distance from the center is the length of this segment, right? Let's just take the coordinates of this point A, X, Y and Z. And let's express in terms of X, Y and Z the lengths of the OA. Well, first of all, we have to drop a perpendicular from the point A to the X, Y, Z. That would be easier, right? And perpendicular from this to this and this. So this will be my AX and this will be my AY. So the lengths of OAX is my X coordinate. The lengths of OAY is my Y coordinate. And obviously the lengths of the AB would be my Z coordinate, right? Because if you will perpendicular to Z it would be something like this. So it will be the same as AB. Now, let's just think about all AB is the right triangle because AB is perpendicular to the plane, XY plane, right? So it's perpendicular to any line including OB. So this is the right angle. So OA is equal to square, is equal to OB square plus AB square. Now, let's talk about OB square. Now, OB square, now this is the right triangle, right? OB AX is the right triangle. Now, all AX we know, that's an X. Now, AXB is obviously the same as OAY which is Y, so this is X. This is Y. This was Z by the way. So OB square is equal to X square plus Y square, right? OB square is equal to OAX square plus AXB square which is X square plus Y square. Now, AB square is Z square. So my OA square is equal to X square plus Y square plus Z square. And what did I say in the very beginning that this is a sphere, right? Sphere set of all the points which are equidistant from the center at the distance equal to radius, right? So this is supposed to be equal to R square. Now, and this is an equation which defines a sphere in the three-dimensional world with a center at the origin and the radius R. So my point was that using the coordinate system you can always represent algebraically certain things which you might or might not want to consider geometrically and in many cases it's very useful not only to define your position in space but also to research certain properties of certain complicated geometrical figures which you might not actually be able to research in any way other than algebraically. And that's the very important purpose of the coordinates. All right, so basically that's all I wanted to talk about. And as I was saying, Cartesian coordinates is something which you are, I'm sure, familiar with but at the same time I was trying to present maybe a little bit more mathematically a little bit more rigorously the same topic. And plus examples like this are very, very useful. Whenever you have some geometrical object in many cases it's very interesting and useful to represent it algebraically using, in this case, Cartesian coordinates and we will talk about other coordinate systems as well. Well, that's it for today. Thank you very much and good luck.