 Hi, I'm Zor. Welcome to Unizor Education. After we spend some time introducing the concept of a vector, well, let's try to analyze what exactly we can do with this, what its characteristics, etc. So today's lecture will be about one of two main characteristics of the vector. The main characteristics are magnitude and the direction. So today we'll talk about magnitude and considering that the vectors represent certain physical or geometrical quality of some object, we can talk about the lengths of the vector basically. So the magnitude as a philosophical concept is represented in the lengths of the vector. Now if vector is represented graphically like this particular arrowheaded segment, the lengths must be measured in certain units. So for instance this particular vector represents a speed. Well, speed can be measured in meters per second or miles per hour or whatever else. So somewhere we should have a unit of lengths which is equal to let's say one meter per second and then depending on the lengths in these units this geometric representation of the vector has certain lengths like 5.7 for instance. It means the speed is 5.7 meters per second. That's what exactly this represents. Now this is kind of trivial and easy. Now let's talk about the analytical representation of the vectors as as tuples as ordered set of numbers which are related to coordinate system. Let's say we are talking about two-dimensional vectors. So this vector has certain coordinates AB and the tuple A comma B represents the vector because it represents the endpoint of a segment which is originated at point zero zero obviously and so this particular segment represents both the magnitude which is represented in its lengths and direction which is represented basically by the direction from the point zero to point A comma B on the coordinate plane. Now considering we know only AB about this particular vector we don't have it visually, geometrically, we don't have anything. We still have to determine its lengths. Well but let's think about what is a coordinate system. Coordinate system is something which also has certain units of measurement on each on each axis. So the unit of measurement is given because the coordinate system is given. And what are these numbers A and B? Well these are the lengths of the projections of our point, the endpoint of the vector, to the corresponding axis which make up the coordinate axis. So this particular segment which is a projection onto the x axis is equal to three point something and this particular projection on the point on y axis is one point something. So what we know is maybe we don't know the lengths of this particular segment but we do know it's two projections on two orthogonal axes. Now obviously we should use the theory Pythagorean theorem to find the lengths, right? So this piece is equal to A, this piece is equal to B. So the lengths is equal to square root of A square plus B square. Because this is the importance and these are two categories. So what I want to do, what I want to actually say that this is the formula for the lengths, well this is a two-dimensional case. Well let's just spread it around. What if it's one-dimensional vector? Well one-dimensional vector is a vector which is represented on one particular single one-dimensional line and again there are units of measurements here and vector is again from zero to some point let's say to this point or to this point doesn't really matter. So in this in this particular case the coordinate itself but in this case it's 2.7. If the vector is directed towards the negative side it will be minus something. So what would be the lengths? Well the absolute value of this number, right? So in one-dimensional case length is equal to absolute value of the number which represents this is a one topo, so to speak, one-dimensional vector in its analytic representation. By the way these two formulas are not really different because you can always put it like this. This is the same thing, right? So if one coordinate, its square root of this one coordinate, if it's two coordinates it's square root of sum of two coordinates and now obviously we will go to three coordinates and you will see that things is very much similar. The vector in the three-dimensional space this is x, this is y, this is z. This is our point where the vector ends. Now this point can be projected onto the x, y plane. So if this is a, b, c then this will be a, this will be b and and this would be c. So imagine x, 0, y is a horizontal plane. Now the z-axis goes upwards so from this point I'm projecting to the x, y plane so this point is a projection of this point here. Now what's the coordinates of this point by the way? a, b, 0, right? And this height from the point itself to its projection on the x, y plane is actually the third coordinate. So this is c, this is the same length. So again this piece is equal to b, this piece is equal to a which is this one and this piece is equal to c. So what's the length of oa, oa. Now the projection is b and projection of b to these two are let's say c and g. Now oa is a hypotenuse of a triangle a, b, o. So a is above the plane, b is its projection so ob would be perpendicular to, so this is the right angle. So oa square is equal to ab square plus ob square. Now ob square, now this is the hypotenuse and this angle is the right angle, right? Because this is the projection onto this axis. So ab square plus instead of ob square, again Pythagorean theorem, I will use oc square plus bc square plus this. So this angle is right angle. If you look at this from the top, right? It's a projection which means we're directing the perpendicular. And now what is this? ab square is basically c square, the third coordinate. Now oc, oc is our a square and bc same as ob is b square. So again we look at this and we see exactly the same thing that the length of the vector is equal to square root of a square plus b square plus c square. So this is a typical situation with all the vectors regardless of the dimensionality. With one vector, with one tuple representing a one-dimensional vector, we had a square root of its one and only coordinate. In two cases, in two-dimensional case, the two tuple, ab. So the length is square root of a square plus b square. So again it's sum of squares of coordinates. And in three-dimensional case exactly the same thing. So basically using this, using this consideration, we can expand the definition of the length of the vector to any dimensionality. Now I mentioned to you that vector can for instance represent the oil refinery for instance, where all the parameters which are controlling the oil refineries are basically numerical components of this 100-dimensional tuple representing the state of the refinery. So the vector can be in 100 dimensions, right? So how to calculate the magnitude of this? Well, to tell you the truth, in case of oil refinery, it's very difficult to think about meaning of the vector, meaning of the length rather of the vector and direction. I mean this space is not uniform. I mean one component can be let's say a pressure and another a temperature. How can you combine together pressure and temperature? It's really hard to do. However, in some cases when components are not so drastically different, then we probably can't think about the meaning of the of the of the length of the vector. But in any case from the mathematical standpoint, which is a pure abstraction, we can always introduce something like this. So if vector has the components, we can always think about the lengths which is equal to the square root of some of their squares. So in this case it does make sense. So basically this is the definition of the lengths for all these strange situations. But in one, two, and three dimensions, it's quite an obvious thing that the formula is exactly what it is. And that's why it was expanded to other dimensions as well. Well, okay, so right now we basically know what the length is regardless of the way how we represent the vector. And we know about the geometric representation as just, you know, arrowheaded segment with a certain unit of measurement. You can just measure the lengths of the segment geometrically. Or in case it's represented as a tuple, then something like this would be the formula to to represent the the lengths of the vector, which means we know what is the magnitude of this particular vector. What is the magnitude of the of the certain physical or a real object this vector represents, whether it's a speed or a force or anything like this. This is the strength of the force or the real speed of the velocity, something like this. Okay, so this is a preliminary lecture about the lengths. Probably there are certain problems which I will think about and if I will come up with interesting problems I will just, you know, something in as a lecture. Meanwhile, you are obviously invited to take a look at the everything else whatever exists on Unisor, unisor.com and take a little bit more time to register to get engaged in enrollment in certain classes and then you will be able to take exams, which is very useful and you can take any number of exams and any number of times, each exam until you will reach the perfection. So thanks very much and good luck.