 Hi, I'm Zor. Welcome to Unisort Education. I would like to continue discussion about how the distance is measured in Cartesian coordinates and this lecture is about the distance between two points on the plane. So you have a plane, well, this particular whiteboard is the plane, and on the plane I have Cartesian coordinates, which means I have two perpendicular to each other lines. On each of these lines, I have Cartesian coordinate system with coinciding origin for both of them. So this point of intersection is origin for both this and this. So this is called x-axis, this is called y-axis, and I have two points, I would like to measure the distance between, right? So let's say two points are somewhere, it doesn't really matter where they are. This is point A with coordinate xA, yA, and this is point B with coordinate xB, yB. So what does it mean that this particular point has coordinates xB and yB? It means if I project, drop the perpendicular to the y-axis, then I would have the point yB. If I will project B towards the x-axis, this would be the point with coordinates xB. Similarly, xA would have xA and here xB. These are coordinates. So every point is projected to both axises, axis and projected means I drop the perpendicular to it. Wherever this perpendicular drops, the point has some coordinate on this line because it has its own coordinate, and this line has its own coordinate. Oh, wait a minute, this is yA. Okay. So that's the picture which I would like to start with. Two points, each one has certain coordinates. Now, the distance between them, the distance between this point and this point, not sure how visible it is. So what do we do? Well, what I suggest to do is to continue these perpendiculars until they intersect the other lines. Now, what do we have? What's interesting is, let's call this point M. And this point M. Well, obviously, since these are perpendicular to the same line, they're parallel, these are perpendicular to this line, they're parallel, all angles are 90 degrees, so NAMB is a rectangle. Now, I'm using certain properties which are explained in more details in the geometry part of this course. So you basically can consider the proof of whatever I'm doing here as based on certain geometrical properties. If you did not study these, well, you can just skip the proof and get the result. But the proof will be obvious after you will study this piece of geometry. So I'm using the geometrical, language and geometrical theorems and properties as basically is explained in the course somewhere else. So, again, AMBM is a rectangle. Now, AB is it's a hypotenuse, well, not hypotenuse, it's diagonal, but it's a hypotenuse in this triangle which is half of this rectangle. So, ABM is a right triangle as well as ABM, and AB is a hypotenuse, which means we can use the Pythagorean theorem to get its length. But now, let's talk about what are the catcher tip of this particular right triangle. Well, the catcher is BM, well, obviously the difference between the distance between B and M is the same as distance between these two points, which have coordinates X, B and X, A. So, since Pythagorean theorem deals with the square of each catcher tip, so I need the square of the distance between B and M which is the same as distance, a square of the distance between X, B and X, A. And we know that that particular thing is X, B minus X, A square. That was explained in the previous lecture where I was talking about the distance between two points on a line where Pythagorean coordinate system is established. Now, very similarly, what about the catcher tip AM? Well, since this catcher tip is equal to exactly the distance between Y, A and Y, B, then the square of the distance between A and M or between Y, A and Y, B is this. And now, let's talk about Pythagorean theorem. Square of this plus square of this is square of the k-partials. So now I have the formula. D square is equal to where D is the distance between A and B, sum of this. Square of the distance between abscissists and edit to square of the distance between coordinates of these two points. This is a general formula. And I told you in the previous lecture that it's the square of the distance which is a little bit more frequently used as a formula for the distance between two points. And now you see that this actually is very much resembling the one-dimensional case on the line. On the line you have this. In two-dimensional case, you have basically the same formula on each dimension. X is one dimension and Y is another dimension and you add them together. Basically, that's it. This is a very simple formula and obviously it's used quite extensively in mathematics. And well, in the space, you will have the third dimension and it will look the same. So we'll talk about this some other time. More than that, the higher mathematics are... Higher mathematics has a subject, basically, where multidimensional spaces are studied and in the multidimensional space, the formula is exactly the same. You have to have the square of the distance on each dimension and then you add them together to get the square of the distance between two multidimensional points. All right, so I would suggest you to read the game, the notes for this lecture. They are on Unisor.com. And well, that's it. Thanks very much and good luck.