 Hi, I'm Zor. Welcome to Unizor Education. I would like to complete the overview of the vectors before I go into the details of operations, etc., with one lecture about what is equal, what is equality, what is congruence in the vectors world. Let me first start from intuitive geometric representation of the vectors. Now, we are talking about some space where the vectors operate and let's say this is the plane, this whiteboard will be the plane and I have a point which is moving on the whiteboard and it moves with certain velocity which I represent with this vector. Now, let's talk about another point here which is moving towards the same direction and with the same absolute value of the speed. Can I say that these two vectors are equal? Well, in many respects, yes, because let's think about what is a direction. The direction is, in both cases, the same. It goes this way. Now, what's the magnitude? The magnitude is the length of these two segments, which is also the same. So, I think it makes absolute sense to say that in geometric representation, two vectors on the plane which are originated at two different points, but point to the same direction meaning that these are two parallel lines and the segments which represent the lengths of the vector are exactly the same or again congruent, if you wish. Then we can call these two vectors congruent. Now, in the old days the term was actually equal but right now it's more in both to use the congruence as a term. So, basically, my point is that two parallel vectors from two different original points but having exactly the same lengths are congruent. Now, congruence in geometry usually is related to some process of transformation. Let's say this triangle is congruent to this triangle because I can reflect it and shift it a little bit and they will coincide, right? Now, two circles of the same radius are congruent because I can parallel shift one and it will coincide with another. So, this superposition is important for congruence in geometry. Now, can I superimpose these two vectors so they will coincide with each other and what kind of transformation should they use? Well, I obviously can use a parallel shift because if I will connect these two lines, now what I know about these two are parallel and of the same length which means this is parallelogram so these are parallel lines as well. So, I'm shifting parallel to itself this vector towards this and they will coincide. Now, what if I will have the situation when I have one vector and another vector like this of the same length? Well, I can actually superpose one into another using the rotation, right? Now, with vectors it's not allowed. Why? Because direction is the most important characteristic. One of the two actually only characteristics magnitude and direction. So, if magnitude is the same but direction are different, I cannot really superimpose one into another because it would distort my direction. So, rotation is not allowed and so not allowed is reflection. I mean, I can reflect one vector over another and they would coincide relative to this axis but that would distort my direction because the direction was opposite. So, I cannot use rotation, I cannot use reflection, I can only use the parallel shift which preserves both direction and the magnitude. Reflection and rotation do not preserve direction. Now, obviously if two vectors even parallel to each other but they have different lengths, obviously I cannot superimpose one into another so they would coincide. So, only vectors which are parallel to each other and having the same lengths are congruent. Now, if this is true, then I don't really have to connect my vector to the point where it's physically originated. Let's say this point is moving on the plane and it's tempting actually to put the vector of velocity exactly at the point where this point is located at the time. But at the same time, I can put it somewhere else. For instance, if I introduce the system of coordinates, I can always position this particular vector originated at point zero zero directed parallel to this one and having exactly the same lengths and I will say, well, these two vectors are congruence so I will analyze this vector. And I know how to analyze numerically this vector because I can always find out what's the endpoint coordinates and analyze it numerically. Now, if I have two vectors which are congruent to each other in geometric sense, obviously they will be congruent to exactly the same vector which is originated at zero zero. So the congruence in geometrical sense means parallel and similarly directed, the congruence in tuple implementation of the vectors when just a couple of numbers in case of two-dimensional space or three numbers in case of three-dimensional space represent the vector. All I have to do is just check if my representation is exactly the same because representation of this vector will be AB, but representation of this vector will also be AB. So two vectors congruent to each other if and only if their components are correspondingly equal. So pairs are equal if each individual component, corresponding components, first to first, second to second are equal. If it's a tree tuple, like a vector in space, then I need three equalities between the coordinates to make the triplets equal to each other. And if I'm talking about the oil refinery vector of change which is let's say 100 parameters, 100 tuple, then I need 100 equalities among corresponding coordinates or numbers which represent this vector to make sure that two vectors are exactly the same. So if I have two oil refiners, let's say, and I'm asking, okay, are they changing in exactly the same fashion? I have to check each and every parameter if it's changing by exactly the same value towards exactly the same direction, plus or minus, increasing or decreasing, right? So the condition on geometrical representation of congruence of two vectors is parallel and similarly directed in the same absolute, where are the same lengths? The condition on tuple representation of vectors is that every corresponding component must be equal. This is a number and this is a number. Numbers must be equal to each other. Well, that's basically it. Why did I do it? Primarily because I don't want to deal with these vectors in geometric representation. I want to deal only with vectors which are originated at zero. It makes much, much easier because if I will take two vectors originated at zero, I can manipulate them, let's say, add them, subtract them. We will talk about all these manipulations and as a result, I will have exactly the same vector of the same type will be originated in the beginning of the coordinate origin. Now with these two vectors, it's more difficult to deal with them because first of all, if I want to introduce these vectors, I need coordinates of the beginning and the coordinates of the end. So it's like two pairs or two triplets depending on the space, dimensionality of the space. So I prefer to deal with only one end having another end fixed in geometric representation and that's what enables me to represent the vector as a table which has only the number of parameters which corresponds to the dimension of the space, two in this particular case. In this case, I need two and two and this two and two. And now I have to check. Now, let me just give a concrete example. If this vector has coordinates, let's say, I don't know, let's say three, five, and this is two, four. Now, this is, I don't know, something like six, six, and this is three, five. Are these two vectors congruent or not? Well, that's not easy to determine, let me tell you. But if I'm comparing only these vectors which are originating at the same point, all you need to do is just to compare the numbers which represent the end. They must be equal. So these two might or might not actually, I can't even calculate right now very fast, to be parallel and having the same lengths. I mean, I have to use some Pythagorean theorem, et cetera, et cetera. Too difficult. This is much easier. And since these two are congruent and these two are congruent, so that's how I can compare their basic representation as vectors originated at zero, zero. And that's much easier than to compare their spatial representation with origin not really fixed to any point. So from now on, when we are talking about geometric representation, we would try, at least not always, but we would try to use this particular representation with vectors originated at zero, zero. And if we are talking about topical representation, which has only the numbers which represent the coordinates of the end point, then obviously it's assumed that this is a vector with a region at zero, zero, zero. So this particular vector is this one. If this is the point 3, 5, then this is the vector 3, 5. All this. Topical representation implies the, this, I don't know, centrally located or whatever you want, with origin at the zero, zero point. All right. So that's it for this particular lecture. I wanted to make sure you understand that the congruence is extremely important with vectors. We have to know when things are equal to each other, right? And the congruence is defined analytically very easily. The coordinates must be just equal to each other. Geometrically, you have to really imply this congruence with parallelism and unidirectional analogy. All right, that's it for this short lecture. I think that would complete my, the information which I would consider to be an overview about vectors, basically, what they are, what they mean, et cetera. We will have much more, obviously, with operations on vectors, manipulations with vectors, how to multiply, add, et cetera. Some physical, maybe, applications will be discussed as well. But that's in the future. So far, this completes the overview. Thank you very much and good luck.