 Hi, I'm Zor. Welcome to a new Zor education. I would like to start the new topic, the vectors. Well, I am pretty sure that everybody actually heard the term vector. Okay, what's the vector? That's what it is, right? Well, yes and no. As usual, everything is much more complex than it seems to be from the beginning. So I would like to start from some philosophical concepts. We know that mathematics is basically an abstraction. It's all about abstractions. For instance, the concept of a number is an abstraction from quantity, basically. So people do understand what quantity is and to represent the quantity they invent numbers. And also numbers can be used to represent lengths, distance, and some other things. Now, there are other aspects of our life or science which are very important, and they need this abstraction to research, to deal with this mathematically. And this very important concept which I would like to present to you is the concept of direction. So this arrow actually represents some kind of a direction. So the concept of direction is very important. And not only from the abstract standpoint in mathematics, it actually plays extremely important role in science and physics and chemistry. Well, even in control of some browsers like oil refinery, whatever, it's really very, very important. So mathematicians needed to abstract out this particular concept, and they did it. And the vector is actually an abstraction which includes in itself the concept of direction. But it also includes the concept of quantity. So if number is an abstraction from quantity, vector is actually an abstraction from both quantity and direction. So if you take an object which has these two properties, the quantity and direction, then you can actually represent this object mathematically as a vector. Now, the elementary example is a point which is moving somewhere. And the speed, the absolute value, how fast it's moving is represented by a number, let's say miles per hour or kilometers per seconds. But the direction is represented somehow. And that somehow is actually what I'm going to talk about right now. So if you combine the distance and direction of movement, you can have something which can be called displacement. So if you combine the speed of movement, the absolute value of speed and direction, you can have another interesting term, which is velocity. So velocity is basically a vector which combines itself with the quantitative side of the movement, how fast you move, how many meters per second or whatever, and the directional side of the movement, so where you are moving. OK. There is one important concept related to direction, which I would like to address right now. What's important is not only just direction as a term by itself, it's important where exactly you are moving. So you can move, for instance, along the line, or you can move on the plane, which is represented by this white board, or you can move in space like a rocket which moves to the moon. Or movement can actually be much more complex than that and in different spaces. So basically, the concept of a space is something which I would like to bring to your attention. Space is where the movement takes place, or if you wish, where exactly this directional characteristic of a vector is located. Now, space is very important. And I have already just gave you three examples of the space. One space is the line, and the vector on the line should represent, again, two things, like quantitative and directional characteristics. Now, for instance, we're talking about a point which is moving along this line. Now, obviously, it has certain speed characteristics, like how many centimeters per second it covers. It might be constant if it moves constantly with the same speed in the same direction, or it can be variable. But at any given moment, we can talk about speed, which is how fast it's moving. But at the same time, we can talk about the direction, where it's moving, to the right or to the left. Or if we introduce the coordinate system, that's the point moves towards increasing the coordinates or decreasing the coordinates. So how can we represent this particular vector, which combines the quantitative part, which is how fast it's moving, and directional part, where it's moving? Where, actually, it's very easy. On the line, since the speed of the movement is basically the real number, and it's positive. I mean, if we're talking about just how fast it's moving, it's just a positive number, how many centimeters per second it is covered. But if we will add the sign to this, associating the positive sign, the positive speed with movement towards increasing the coordinates, and negative numbers with decreasing the coordinates, then speed by itself, let's say, 5 centimeters per second, with some kind of a sign, plus or minus, this together characterizes the vector of movement, because it has an absolute value, how fast it's moving, and the direction. In this particular case, it moves covering 5 centimeters per second towards the increasing the coordinates. So that's the representation of the vector on this particular line. Now, can it be represented in more graphical form? And I will definitely talk about this in other lecture. Yes, absolutely. It's basically very easy. What you can do is you can draw an arrow here, which ends up at the point 5, and say, OK, this particular arrow represents my vector. It has nothing to do with the position of the point. Position of the point can be anywhere on the line. But the velocity, the speed and direction of its movement can always be represented as the arrow which starts at 0 and ends where exactly the numerical characteristic ends up with a proper sign, plus 5 or minus 3 or whatever else. So we're talking about numerical and the graphical representation. But I was talking about space, right? So in this particular case, we're talking about one-dimensional space. Now, why is it one-dimensional? Because if we introduce the coordinate system, it's enough to have one real number to represent a position. And if something is moving in this particular space, then the velocity of this movement can be represented also on the same line using either the arrow or some kind of a number. So position and the vector itself are represented in a very similar way by one real number. Position can be anywhere, and the velocity can be basically any real number. Now, let's talk about some other spaces. Well, the next from the complexity standpoint is the two-dimensional space. So let's consider we have a point which is moving somewhere on this whiteboard. Again, what we can do is we can always draw some kind of an arrow saying, OK, well, that arrow represents the velocity of this particular line. The lines of the arrow represents the value of the speed itself, the magnitude of the speed, the absolute value. And the direction represents, well, the direction where exactly it moves. Now, in the coordinate system, if we introduce some coordinate system, the position of the point is always two numbers, right, abscissa and coordinate. Now, if our point moves towards this direction, then I can always use this particular arrow, putting it in the beginning of the coordinate, originate it from the origin of the coordinate, and say that this arrow represents the speed, because it represents the direction. It's the same direction, and it represents the quantitative characteristic, the magnitude of the speed. Now, going into a little bit more complex case, let's say you control some oil refinery or something like this. There are many, many parameters which are changing all the time. If you just look at the, probably you have some kind of a dashboard or whatever, where all these parameters are somehow brought together through some gadgets, widgets, et cetera. Now, a state of this oil refinery is a set of all these parameters taking a particular value at a particular time. And then the state is changing. So you are talking about hundreds of different parameters changing at the same time. Now, here we have only two parameters, abscissa and ordinate. But in case of an oil refinery, you have hundreds of parameters. Now, they do actually represent the movement. But in this case, it's not two-dimensional space. It's the dimension which is basically equal to the number of parameters. Let's say you have 100 parameters. So it's a 100-dimensional space. And detectors do exist in this space as well. Because since our oil refinery is changing, all the parameters are moving somewhere, so the point which represents a position, a state, if you wish, of our oil refinery at any given moment, this point has 100 coordinates. And it's changing these coordinates. So the point is moving in the 100-dimensional co-ordinate space. And we have to represent the vectors over there as well. Because we have to control this particular oil refinery. So all the parameters should be within certain limits. They shouldn't change very fast. The temperature cannot rise very fast or fall very fast or something like this. And same thing with, I don't know, something else. And if some parameter, let's say, falls faster than it should, well, this is something which we have to really pay attention to. So that's why it's very important to understand that together with the concept of vector, which includes in itself the magnitude and the direction of movement, you really have to understand the concept of a space where this movement takes place. And as a final example, before I complete this introductory lecture about vectors, it's about special theory of relativity, Einstein's creation, where the space where things take place is four-dimensional. It's three dimensions of our coordinate system, Euclidean coordinate system plus time. So now you have four parameters which characterize the position of something in this world, time and three spatial parameters. And there is a special geometry there and there are vectors over there. I mean, these parameters are changing. Time is changing as well as the spatial coordinates. So this is a perfect example of a four-dimensional space where vectors exist and they must be researched somehow, et cetera. Vectors are very convenient in physics. Many, many processes, not only the mechanical movement is represented as vectors in physics. For instance, the force. Force is basically a vector because it has a magnitude and direction where it moves. Like the force of gravity, for instance, in this particular point depends basically on two things, the mass of the Earth, the mass of the subject, which is basically gravitated towards the Earth, and the direction. The direction is vertical down. So the vector, which represents the force of the gravitation of the Earth, is basically something like this arrow or you can represent it in many other ways. But for instance, it's an arrow. The length of the arrow is the magnitude of the force and direction is down. Now, if you are talking about some kind of a spot in a solar system where you have the gravity of the sun and planets and some other formation in the space, then it's not so obvious where the vector is basically directed. And it's probably changing because the time goes by. You circle around the sun and your vector is changing. And even in this elementary example of my gravitation towards the Earth, well, in a big picture considering the space we are in, as the Earth is rotating, obviously this vector is also rotating together with the rotation of the Earth. So basically what I would like to say that vector algebra, as it's sometimes called, is a very important part of the mathematics and very useful for many practical purposes or science purposes, research, et cetera. So that's it for today. I am planning to have more information, more general introductory style information about vectors and their representation. And then we'll probably devote a lot of time, basically, operations on vectors and as much as we dealt with operations and numbers like multiplication, addition, et cetera. I mean, we will have some similar things in the vector algebra as well. Well, that's it for today. Good luck.