 Hi, I'm Zor. Welcome to a new Zor Education. I would like to continue this overview of vectors. It's not really the real research or real theory. It's just basically illustration of what vectors are and how to deal with them. And let me just recall that we were using graphs to basically understand better how functions behave. Similarly, the geometrical representation of vectors is used more for illustration purposes rather than for analytic calculations or whatever else. So today I will concentrate primarily on geometrical representation of vectors. So we know that vectors are objects which have two major characteristics, the magnitude and the direction. It's very important to understand that if we are talking about direction, we have to really think about certain space where this direction is pointing to. If we are talking about vectors as representing, for instance, velocity at the moving point, then this moving as action occurs in certain space. If we are talking about vectors as the representation of the force, for instance, gravitational force, again, it's applied to certain objects which are located in space. So the direction as a concept is always related to the concept of a space. So what kind of spaces where things are moving or forces are acting can be talked about and how can we interpret geometrically or graphically, if you wish, the vectors in these spaces. Well, let's just consider a few examples. Let's first think about a point which is moving along the straight line. So this is the straight line. The point is moving. So this is the point. And we would like to express in the graphical form the vector of its velocity. Okay, velocity has an absolute value, speed, how fast the point is moving, and the direction. Well, there are only two directions, right? So basically the typical representation which is used in this particular case can be just an arrow which has the length, this is the length of the arrow, equal in some units of measurements to the magnitude of the velocity of this movement. And the direction, this arrow at the very end, represents the direction along the line where actually the movement occurs. Now, what if a point moves not along the straight line but along some kind of a curve? Can we represent the velocity as a vector in this particular case? Well, yes, because if the point is positioned, let's say, here and moves this way along the curve, what usually is used is a tangential line, tangential in this particular point. And again, it's an arrow which has the length equal to the absolute speed the point is moving and the direction is towards this side of the curve where the movement actually occurs. Okay, now, these two cases represent the so-called one-dimensional spaces. So straight lines and curves, they are one-dimensional. Now, why are they called one-dimensional? Primarily because we can introduce the system of coordinates, which means we have to introduce the beginning of the coordinates. Let's say this is point zero and the unit of measurement. And if we will introduce these two things, then every point on this particular straight line can be characterized by one and only one real number, which basically is the distance from the point zero in these units positive on this side and negative on that side. Now, with the curve, the situation is exactly equivalent. We can always find some kind of a point zero. And again, using this particular unit of measurement, we measure one, two, three, four, minus one, minus two, et cetera. So point on the curve will also be defined by some real number. Of course, I'm not touching more complicated issues about what is the length of the curve, which is not defined. I perfectly understand that this is something like a higher levels of mathematics. But intuitively, you obviously understand that if the curve is smooth enough, then the lengths can be defined, as well as tangential line. I was talking about tangent to a curve. And again, not every curve has a tangent in every point. But we're talking about smooth cases when tangents do exist. And in these cases, we can talk about representation of the velocity as a vector. Now, how about two-dimensional cases? Well, two-dimensional case, first of all, is obviously a plane. Let's say the surface of this white board is a plane. And again, I can introduce the system of coordinates here. And obviously, position of every point is defined by two coordinates, subsistent, originate. Okay, fine. Now, what if the point is moving? Well, when it's moving, again, it's moving on the surface, on this plane. And at any point, we can always have some kind of an arrow which represents a vector of the velocity of this movement. Or if there is some kind of gravitational force which is directed down, that also can be represented as an arrow in this particular direction down. And the magnitude of the gravitational force is reflected in the length of this arrow. So that's how we can graphically represent the vectors in the two-dimensional space. A little bit more interesting example of the two-dimensional space is, let's say, a sphere, for instance, or the surface of our planet, which, you know, roughly can be considered as a sphere. Now, it's not a plane, obviously. However, it's also a two-dimensional surface. Like in case, for instance, of our planet, you know that we have longitude and latitude. These two coordinates define the position of every point on Earth. So, again, since two numbers needed to determine the position of the point, the surface is called two-dimensional. Now, what's interesting is that, well, it's not as obvious in case of a plane, but in case of, let's say, a spherical surface. And there is a point moving, let's say, this is an equator, and the point is moving along the equator. Also, the vector would be tangential to the trajectory of the movement. So, wherever the point is moving, whether it's on a sphere or on a plane or any other two-dimensional surface, the trajectory on this particular surface will be some kind of a curve. And the vector which represents the velocity of the point moving along this trajectory will be a vector, will be an arrow which is tangential to the curve. All right. Three-dimensional space. Well, obviously, we have vectors in three-dimensional space as well. Three-dimensional space is the one which we live in, or at least we thought. We used to think that we live in a three-dimensional space until Einstein disillusioned us with theory of relativity, adding the time as another coordinate. So, this is x, this is y, this is z, abscissa, coordinate, and I was going to call it applicant. Right, the third coordinate. The position of every point is defined by three coordinates, x, y, and z. Now, again, if this is a point which is moving along certain trajectory in space, we can always think about a vector represented as an arrow which points towards its tangential to the curve in a three-dimensional space. Now, what's the example of a curve in three-dimensional space? Let's say a spiral, right? Calyx or something. So, it's tangential to the trajectory, but it's a three-dimensional vector, which means this particular arrow has a direction and a length, but it's all in 3G in space, like from here to there. What's also interesting is that how this particular thing is changing with the time. Well, if the point is moving uniformly along this particular curve, uniformly in terms of constant speed, the length of this vector is not changing, only direction is changing, right? Now, if this is a straight line somewhere in space or on a plane or wherever, and it's a uniform movement, then vector actually is exactly the same. The beginning of the arrow is moving along the straight line, but the length of the arrow is the same and direction of the arrow is the same. So, all these vectors are actually the same. And here is something which I would like to point out. We are talking about vector as something which has a magnitude and a direction, which means if two vectors have the same magnitude and the same direction, then they are actually equal, right? So, if we will have, let's say, on the plane, and let's say we have coordinates even, if we have this vector represented as an arrow which has certain magnitude and certain direction, and this arrow, which has exactly the same direction, it's parallel and the magnitude is the same. Well, then these two vectors we are saying they are equal to each other, or congruent, whatever the term you prefer. So, that's why we could always consider only the vectors as errors originated in the beginning of the coordinates. So, if point is moving along some kind of a curve, so this is another vector. What I can say is that this vector which is the same as this and this vector which is, let's say, the same as this are two different vectors which reflect the velocity of this particular point at two different times. So, the velocity is changing from this to this, and it's exactly the same as I'm saying that the velocity is changing from this to this, because these two vectors are considered to be completely equal or congruent to each other. So, that's why the manipulation of these vectors is always more convenient if you bring all the vectors to the beginning of the coordinates. Later on we will talk about like addition or subtraction of a vector or multiplication of the vector or something. And in these cases it's very convenient to forget about the origination point of this particular arrow. It's more convenient to originate them all from point zero because these are exactly the same vectors and that's why the result of the operation will be exactly the same. What else I wanted to talk about? Geometrical representation. Well, basically that covers more or less. The most likely examples which we will be dealing with will be about vectors in a two-dimensional space which I will use the whiteboard actually as the graphical representation. But again, don't forget that this is just a graphical representation of these arrows. It's just to illustrate better how to deal with vectors, how to add them, how to subtract them, etc. The analytical representation of vectors using the system of coordinates which will be covered in the next lecture is kind of more useful, more precise, and that's the real science, the real mass, if you wish. This again is no more than illustration. Inasmuch as graphs are illustrations to behavior of the function. It's very convenient to see it this way. Now, also should be mentioned that vectors are something which really is very much used in physics, in different areas of science, etc. Well, that's basically it. These are still introductory lectures about vectors. The real theory will be further on when I will finish with all the overview kind of things. Next lecture will be about analytical representation of the vectors. And only then I will start thinking about a little bit more theoretical lectures with problems, examples, etc. Well, thanks very much for today's lecture and good luck.