 Hi, I'm Zor. Welcome to the New Zor Education. We're talking about vectors and their representation. It's still part of the overview about vectors. So after introduction of the concept of vectors and a certain time devoted to geometrical interpretation of the vectors as basically directed segments in some space on the straight line or on the plane or in a three-dimensional space or anything. So after that, well, I have to say that, well, the picture is worth a thousand words, of course, but mathematicians do require certain precision and pictures are used for illustrative purposes. It's like graphs for functions. We can use the graphs to basically understand how the function behaves, but concrete analysis of the function is probably possible without formulas, like y is equal to square root of x. I mean, it's nice to have a graph of it, but we do need a formula and the way to calculate certain things to deal effectively with this. So same thing with vectors. They need some numerical representation. Here's how it's approached. Well, let's consider for simplicity the movement of the point along the straight line. So here is the straight line. Obviously, whenever we're talking about some numerical representation of physical movement, we do need coordinates. So we do have zero point, and then 1, 2, minus 1, minus 2, et cetera. Now, here is the point somewhere, and it has certain velocity moving on this particular straight line. Let's say the direction of movement is towards increasing of the coordinates. There is a certain numerical characteristic of this speed which is positive real number, whatever the number is. Now, we can always represent this particular velocity with a signed real number where the absolute value is equal to the speed itself, and the sign corresponds to the direction. If direction is towards increasing of the coordinates, then the sign would be positive, and otherwise it would be negative. So one real number would represent a velocity, not just the speed. Speed is represented by absolute value of this number, but the velocity is represented by both absolute value and a sign which is an entire real number. So velocity of let's say plus five means that in every unit of time this particular object is moving five units of space towards the increasing of the coordinates, and the velocity minus five means that with the same speed of five units per unit of time, the five units of plants per unit of time, it moves towards decreasing of the coordinates. So one real number, and what's interesting is we are talking about the straight line where the position is also determined by one real number, whether it's on this side of the zero, that side positive or negative. So there is some correspondence between how we determine the position of the point, and how we determine the speed of the point. Both can be represented as real numbers. Of course, they have nothing to do with each other. Position can be plus 25, and the speed can be minus five, which means from the position 25, it moves to the left on this picture with speed of five units of time of lengths per unit of time, which means after one unit of time it will be 20, another unit of time it will be 15, then 10, et cetera, five per unit of time. But in any case, what's important is the correspondence of the dimensionality. One dimensional line and we need one real number to represent position on this line and the speed on this particular line. That's very important. Now let's go to the two-dimensional case. Well, in two-dimensional case, we have two coordinates, obstinate and ordinate, and every point is defined by its two coordinates. Now, let's consider this is a ball rolling freely on the plane, along the straight line with a constant speed, and geometrically, you can always express it like this, where this arrow represents a direction, and the lengths of this segment represents the speed. Obviously, the faster it moves, the longer will be this particular geometrical representation of the vector of velocity. Now, can it be numerically represented? Absolutely. What we need is to determine the direction and the magnitude of this vector, because that's what defines the vector. The direction is this one, but that's the same thing as this one. If these lines are parallel, the direction is the same, this direction. And the lengths I will use exactly the same. Now I will use coordinates of this point and say, okay, the vector of velocity of this particular ball, this particular point, is a, b. So two real numbers which determine the endpoint of the vector, which is exactly the same by its magnitude, and pointing to the same direction as the velocity of this particular ball, this endpoint, the coordinates of this endpoint define the vector. So vector is defined by two real numbers. By the way, again, correspondence, two-dimensional plane, two real numbers to define the position, and two real numbers to define the velocity. So what I'm saying basically is, now, okay, let's go to a three-dimensional case just to make the whole thing complete. In the three-dimensional case, let's say it's a rocket, which is just flying towards the Mars by inertia, just no engines working, et cetera. So it's straight line in the three-dimensional space. It's exactly the same thing. I can go to the beginning of the coordinates, wherever the coordinates are, maybe they're centered at some sun or some star, whatever the coordinates are. Go to the beginning of the coordinates in this three-dimensional coordinate space, and I will draw a vector which has the same direction as our rocket is flying, and the length of the vector in some units of measurements corresponds to the speed of this particular rocket. And I will say that this particular vector represents the movement, the velocity, and its end point, which in this case will have three coordinates, abscissa-ordinate and applicate, will determine the vector. Three-dimensional case, three coordinates for position, and three numbers, three real numbers, which represent the vector of velocity in this case. The point I want to make is that there is this correspondence between the dimensionality of the space, vectors are operate, and the number of real numbers, which are sufficient to determine the magnitude and direction of the vector, which basically define the vector. And now let's go to a completely different case. Let's say you're operating in... you're operating, let's say, oil refinery. Now, this is a very complex facility, and it has, let's say, 100 parameters, which basically determine how everything works. Parameters of every particular component of this oil refinery, let's consider there is 100 of them. Now, every one of these parameters obviously is changing as the time is changing. Obviously within certain normal limits. Now, if we would like to somehow determine this change in the parameters, well, in the language of mathematics, it's a 100-dimensional space, so to speak, because it takes 100 parameters to describe the state of the whole system. And every parameter is some real number, so we need 100 real numbers to define the state of the system. But how about vector? Well, actually, it's very, very similar to this one, because what we can do is we can check how much each parameter is changed during the unit of time. And that's also some kind of a real number. Now, the set of all these real numbers, and there are 100 of them, of course, because there are 100 parameters, so there are 100 changes, these real numbers would characterize the magnitude and the direction where exactly the parameters are moving at any given time. So, it's completely different, not geometrical and not even represented as a vector, a geometrical vector kind of a situation. But still, you can use an ordered set of numbers, not ordered because, obviously, the parameters are meaningful. I mean, the first one is the pressure, the second one is the temperature, et cetera, et cetera. So it's ordered set of numbers, which define the vector. And in this case, it's a vector of change of parameters. In all the cases I was talking about, this ordered set of numbers, the real numbers, actually describes or defines or represents, whatever you wish, a vector. And the number of these parameters, number of these real numbers which define the vector, correspond exactly to the dimensionality of the space this vector operates. So, right now we can forget about graphical representation of the vectors as segments with an arrow somewhere in real space, one or two or three dimensional. We can't define the vector in this case for a hundred dimensional oil refiner, right? But we can instead say that any set, any ordered set of real numbers represents a vector in n-dimensional space. Now, if you want to keep the geometric representation of this, you can view this as coordinates in n-dimensional space of the endpoint of the vector, where the beginning of the vector is at the origin of the coordinates. So, let's say we have a three dimensional space and you can have this point. The coordinates are a1, a2, and a3. So, this will be a3. And this particular point is actually representing this vector. So, let's forget about geometric representation and let's think about the ordered, sometimes people are using angle brackets, sometimes the round brackets. In any case, let's use this particular representation of vectors. Now, it's more generic. That's number one. Number two, it's precise because these are numbers and we know what to do with numbers. We can add them, we can multiply them, et cetera. We cannot say exactly the same with segments. I mean, sometimes we can, sometimes we cannot. I mean, there are certain restrictions on the geometric representation. But with numbers, we can do anything we want to. Multiply, divide, add, et cetera, et cetera. And that's what would define the characteristics and operations of the vectors much more precisely. Well, this ordered set of numbers is called tuple in mathematics. So, in this case, we might call it n-tuple, which means the tuple which contains n-components. Obviously, the vectors on the straight line can be defined as one tuple. The vectors in the plane is two tuple. The vectors in the space as three tuple. And the vectors in the Einsteinian theory of relativity where the time also is the coordinate is a fourth tuple. So, that's basically it for this particular representation. Now, why it's important because, as I was saying before, this numerical representation allows us to operate on any dimensional vectors, including the vectors which describe changes in the oil refinery with numbers, which is much easier. It can be computerized, obviously. So, that's the most important purpose. Now, that's basically it. So, there are two representations we were talking about, the graphic representation, which is more for illustration purposes. And I will definitely use graphical representation, the pictures, the geometry of the vectors to illustrate certain properties of the vectors. But at the same time, we should not forget that the precise implementation of vectors are through tuples, through ordered set of numbers. And that's what probably would be used to define operations on the vectors, et cetera. So, I will define it using this, but I will illustrate it using geometric representation. All right, so that's it. Now you know another representation, the tuple representation of the vectors. And, well, there are much more information about vectors which I'm planning to share with you. Thanks very much. And don't forget that Unizor contains all these lectures, so you can re-examine them again. These lectures are not really supplemented with any problems. These are introductory overview kind of things. Problems will be later. Thanks a lot and good luck.