 Hi, I'm Zor. Welcome to the Newscore Education. I will talk about operations on vectors, certain condimental operations. It's basically vector arithmetic. Now, when we talk about arithmetic, we usually have in mind numbers and manipulation with numbers like additions, subtraction, multiplication, and division. That's basically arithmetic. Now, with vectors, we will be dealing under the topic arithmetic, addition and subtraction of vectors. As far as multiplication and division, we will be considering multiplication by a real number and division by a real number, not multiplication vector by vector. This is a separate topic. There are two kinds of multiplication of vector by vector. One is called scalar or that product, and another is called vector or cross product, and that will be separate topics. So, today, this lecture is about multiplication and division by real numbers and addition and subtraction of vectors. Now, vectors are new objects. Mathematicians just came up with this. It's something which combines in itself the lengths and the direction, these two components. Now, the operations, arithmetic operations, on vectors should be defined naturally. So, people who deal with these operations would agree that this is exactly what operations of addition or subtraction or multiplication by 25, are supposed to be. So, this natural approach is very important. So, any kind of a definition should be justified by some common sense. And that's what I will try to explain in this lecture. I will try to put some common sense behind the definition of arithmetic operation. Okay. So, let's start. The first operation which I wanted to define is multiplication of the vector by a natural number, like multiplication by two, multiplication by 25, multiplication by one, for instance. Now, how naturally can it be defined? Well, what I'm suggesting right now, and that's what actually mathematicians decided to define, is very simple thing. If you have a vector, then to multiply it by two, one, 25 or whatever else is, retain the direction of the vector and just make it longer by this factor, by two, by 25, by one, whatever it is. Well, by the way, in particular, multiplication by one means that the lines will stay exactly the same. So, since we retain the direction, so the vector stays the same, which basically implies the rule that multiplication by one doesn't change the vector. Multiplication by two makes it twice as long, so it will be about that long. Multiplication by 25 will be 25 times longer. I consider this a very natural definition. Now, what's also very important is that certain properties of multiplication which we used to have with numbers, for instance, commutative law, associative law will definitely be preserved here. I don't want to prove it right now. I will probably do it as a problem, actually, which means I will ask you to prove it first and then I will explain in lecture. But yes, all these laws are supposed to be held. So, for instance, if you have a vector, I will use the lowercase legend, letter and arrow on the top, and you multiply it by number m, and then the result, now this is m times longer than a originally was, and the result I'm multiplying by n, so it will be even n times more than that. What does it actually mean? It's the vector which has exactly the same direction as a, but its length will be multiplied by m and then multiplied by n. But length is a real number, which means whether we multiply it by m and then n, or by n and then m in any order, so these are supposed to be exactly the same results. So, this is some very, very brief introduction to prove of all these laws, but I don't want to stop any longer. I will introduce a little later this. So, this is multiplication by natural number. It's easy, it's natural, and that's why it's natural. All right, now, how about division? Well, division by natural number, let's just think again. What is the definition of the division if you're dealing with numbers? What does it mean that six divided by three is equal to two? Well, it actually means that two multiplied by three is six, right? So, in case of vectors, if I wanted to divide vector by n, it actually means to find another vector which, if multiplied by n, would give a. Now, obviously, if you take exactly the same direction and just shorten the lengths by the factor of n, so this would be one nth of the original vector, then obviously, multiply by n using whatever the definition I just talked about, how to multiply vector by some natural number, I will get the original a. So, that's why dividing by natural number n is very simple. You just retain the direction and shorten the lengths of the vector by this particular factor. Now, if we can multiply by a natural number and we can divide by a natural number and all these manipulations are with lengths only, which is the real number, obviously, what we can do now, we can multiply vector by any rational number because it actually means, firstly, multiply by m and we know how to do it and then you divide it by m and we know how to do that. And again, since all these manipulations are with lengths only, direction is the same as before, then these are just manipulations with real numbers and we know that we can multiply by a rational number by first multiplying by numerator and then divide it by denominator. So, we covered the rational numbers, positive rational numbers, mind you. All right, how about irrational numbers? Well, irrational number can always be represented as a limit case of rational numbers. We did talk about this. So, for any irrational number, there's always a sequence of rational which is approaching it intimately closely. So, basically, we can introduce the same type of approach. I don't want to do it very rigorously right now, but in theory, if you know how to multiply by rational numbers, approximation of any irrational, like square root of two, for instance, with rational numbers would give you, basically, the definition of the multiplication of the vector by irrational number as a limit case of multiplication by rational numbers. Now, and by the way, all these manipulations, multiplication by rational, irrational, positive numbers, obviously would conform to commutative and associative laws. Again, I don't want to rigorously prove it right now. I will leave it for later for the problems, but it should be really felt by anybody, since we are only manipulating the lengths and length is just the real number. Then all the laws which are applicable to multiplication of the real numbers are applicable here. Okay, next. Next, I would like to spend some time. Now, by the way, I did not cover negative numbers, only positive, okay? Now, well, I have this sequence of introduction which would introduce negative numbers a little later, and I do have some reasons for this. So let's just follow this particular structure. So next what I would like to do is I would like to define how to add two vectors which are directed identically, just have different lengths. Well, again, since these two are identically directed, then we can always bring them to common beginning, let's say beginning of coordinates. This will be one vector, and this will be another vector. Now, they are actually coinciding, so don't pay attention that it's kind of two separate lines. One lies on the top of another. Now, what can be the sum of these? Well, let's just think about, what's the physical meaning, for instance, of a vector? Well, it can be a force. So we have some object, and one force, pressure by my hand, for instance, can be expressed in some kind of a vector, and then some something else would pull it, let's say, with another force towards the same direction. Now, what will be the sum of these forces? Well, you just basically say this is sum. So you have to sum the lengths of one vector and lengths of another vector, and again retain the same direction. So manipulation of these vectors, the addition of these vectors, which have identical direction, should result in the vector, which has identical direction with these two, and the lengths, the magnitude of this vector, should be equal to the real sum of these two lengths. I think it's very natural. Obviously, it conforms to all these commutative and associative laws of addition, because again, they don't really change direction. All vectors, two operands, and the result, the sum of these two are all identically directed. So all we manipulate is the lengths. We add one length to another to get the result. And obviously, it works like the addition of real numbers, basically. So this is a definition of addition of identically directed vectors. Okay, so I'm introducing these operations just step by step, trying to make simple ones first, and then I will complicate the issue. Next, what I would like to talk about is something which I would call a null vector. Now, null vector, as the name implies, has the lengths null. But in this case, it's very difficult to talk about direction, right? Null vector can graphically be represented as a dot, basically. So there is no way you can put this arrow. So this vector has certain length, so you can put some arrow at the end, and that's the direction. Point does not have a direction. So when I'm talking about null vector, I can actually say that this null vector has either no direction, or you can consider it directed to anywhere you want. It doesn't really matter. Now, what's interesting about this? Well, let's add this vector and this vector, the real vector and the null vector. Well, since null vector can take any direction I want, but it has a length zero, well, why don't I say that, okay, in this particular case, if I would like to add it to this particular vector, it has the same direction. And in which case, the result would be the vector in the same direction, and the length's equal to sum of two lengths, right? Now, this is zero, the length is zero. So if I will sum them up, I will get exactly the same length. So the result of addition of a null vector to any vector would be exactly the same vector we are adding it to. So this particular concept null vector and operation of addition is really working exactly like with numbers. You can add zero to any number and the number will not change. Okay, so that's the concept of a null vector. Now, what's next? Next is multiplication by zero. So we knew how to multiply by positive numbers, rational, not irrational, irrational. Now, how about multiplication by zero? Well, I can always say that the multiplication by zero gives me null vector. So this zero is a number, the multiplier. This zero with this little arrow on the top signifies a null vector. So multiplication of any vector by factor zero will result in null vector. Now, it actually completely corresponds to the idea of a null vector and the multiplication. The length is always multiplied, right? So the direction is retained when I'm multiplying and the length is multiplied by the factor. Factor is zero, so the result of this multiplication is vector which has the length of zero. And I don't really care about direction because this is a zero, this is a null vector. So I can say that null vector has exactly the same direction as SI. There is nothing wrong with that. All right, so we know how to multiply vector by zero. The result is null vector, okay? What else? Now, I would like to say the following. I would like to expand my definition of addition from identically directed vectors to directed vector. Let's think about again, what is vectors meaning from the physical sense or mechanical, whatever? Well, mechanically, for instance, it might signify a displacement, movement. This means I moved from this point to this and this means I moved from this point to this. Well, obviously the result of summation of these two should be, I just did not move at all. I was stanging still. Similarly with forces. If you have two forces directed against each other and they're both applied to the same object, they nullify each other and the object stands still. Basically, it doesn't move anywhere. So it's very natural to define the operation of addition of these two vectors which have exactly the same lengths and they are directed opposite to each other. It makes sense to define their sum as, as null vector because they kind of nullify each other, right? So, if I will add a vector with another vector which has opposite direction but the same lengths, I will get zero. So that's what basically means. All right, that's good. Now let's consider a little bit more complicated case. What if this is not the same lengths and I would like to add them together? Well, let's not forget that we would like our laws which we are defining right now to be nice, natural, commutative, associative, et cetera. So what actually might happen in this particular case is the following. I can always represent this vector as a sum of vector which is directed the same direction as this one but the lengths should be equal to this one and then another piece. So it's like a sum of these two vectors, this one plus this one. Now if I will sum them together, now obviously again I'm assuming that whatever we are defining must be associative and commutative, et cetera. So we kind of think about this as these two pieces should nullify each other and only this one would remain, right? Now how to describe all this process in a little bit more mathematical form? Well, very simple. So if you have two vectors which are directed in opposite direction, first of all you determine the result of their sum. The result would be the same as the one which has a greater length which is this in this particular case. Now what will be the lengths of the result? Well, you just have to subtract from this length, subtract this one and then you will get this piece and that's the result of summation of two vectors which are directed opposite to each other. By the way, from the position of just defining certain new concepts, there is a concept which is called collinear vectors. These are vectors which are basically along the parallel lines always which means either they are identically directed or opposite to each other but it's still within the same direction. So vectors which have direction either identical or opposite to each other, they're lying like on the same line or lines are parallel depending where exactly these vectors are positioned in space. So they're called collinear which means they're sharing the same line along which they're both stretching either in the same direction or opposite. All right, so we have completely defined the operation on collinear vectors, right? Next is I would like to return back to multiplication. Now we have completely defined quite well actually everything about positive factors which we multiply the vector. Now how about negative? Well, the first of all, I would like to define multiplication by minus one. Now, multiplication by minus one, I will define as collinear vector but directed opposite to the one to the A, the one which I'm starting from. Well, it's exactly the same as with numbers. If you have five, you multiply five by minus one, you get minus five. So minus five and five are symmetrical. They have the same distance from zero. So they're very much similar to two vectors which are directed opposite to each other but they are collinear. So if I will get this as a definition, then from this I can obviously define multiplication by any negative number, by any negative number k where k is less than zero as multiplication by minus one and then multiplication by absolute value of k, right? So since k is negative, multiplication by minus one and absolute value of k should give me exactly the same thing because minus one times absolute value of k would actually give me the k if k is less than zero, right? So I know, again, using the associativity, commutative law, et cetera, I can always say that, okay, I know how to do this. This is just a vector which is directed opposite to the A and then I know how to multiply by a positive constant, positive real number. It's already defined before. So now this actually completes the multiplication case. So I can multiply by any constant, positive, negative, zero, rational, whatever. All right. What's left? Left is the most interesting part. This is something which usually many teachers just introduced from the very beginning, okay? This is a definition of the addition of two vectors and then they basically present what I'm going to present right now. I did all the explanation before this just to gradually bring you to this particular concept. So now we are talking about two vectors which are not collinear. They do not necessarily have the same lengths but let's consider they are of different direction and different lengths and question is how to define naturally, properly, nicely, if you wish. You can attach you in some other epigets but anyway, how to define the sum of these two vectors. Again, let's go back to physical interpretation of the vector. Let's consider vector represents a displacement. So whenever I'm moving from one point to another, that move, that displacement is characterized by this vector direction and the lengths I moved. And this is another move. Now, what does it mean that I would like to add together two moves? Let's say I move one mile north and then one mile west. Well, first you move one mile north, right? Then you take this vector, attach it at the end. So from here, I put it in such a way that the origin of this vector coincides with the endpoint of the first operand. So this is A, this is B, this is also B. It has exactly the same lengths and exactly the same direction but I position it at the end of the first movement. So what's the result of my movement? Well, that's the displacement which is this. So the rule to add together two vectors which are not necessarily collinear and not necessarily the same lengths is you take one vector. Now at the very end, you attach another vector which is exactly the same directed and the lengths the same as the second operand. And the endpoint of that, together with the original point of the first vector of the A, would give me the lengths and direction of the combined movement. So that's the definition. Now, does this definition is more or less the same in case vectors are collinear? Well, yes, because for instance, B and A are collinear and I will just put the B as a continuation of A. Obviously, their lengths will add together and it completely corresponds to my original definition of the addition of two collinear identically directed vectors. How about opposite direction? Well, it's exactly the same thing because if this is A, this is B. What's the result from the beginning of A to the end of B, which is this particular piece? And that's exactly what I was talking about when I was saying that from the bigger lengths you should subtract the smaller one and the result would be the same, the direction would be the same as the bigger vector. Okay, now how else this can be represented? Well, it can slightly different to be drawn, if you wish, because notice that this is parallelogram. Why? Because this is equal to that. Vectors are equal, means they are equal in lengths and parallel and that's why it's parallelogram. There is actually a theorem about quadrilateral with opposite sides parallel and equal to each other that this is parallelogram. And the result of summation is basically a diagonal. So another way to present exactly the same way is not to move B to the end of A, but instead build a parallelogram on these two which are originated at the same point. So you build the parallelogram and take its diagonal from the same point. This is a rule of parallelogram, which is in physics actually very much used in many, many different places. That's how we add forces together. So if you have for instance two forces acting on the same object at different magnitudes and different directions, what's the resulting force? Well, that's the rule of parallelogram because that's exactly how vectors are added together. So we found how to add two vectors. Okay, what's left? Subtraction, right? How to subtract the vectors? Well, but let's think about it. What does it mean? Well, it means basically this. So, if I want to subtract from A, I want to subtract B. I would like to find out the vector which is if added to B would give me A. Now, so basically obviously this is the vector. So it starts from the endpoint of the B and ends at the endpoint of A. And indeed, if I will add B and C, I will get A, right? Because that's how I define the addition. So subtraction is very easy, as I said. Now, what's interesting, and this is just again some kind of side note, we would like to have certain arithmetic rules which we used to have to be true for vectors as well. Remember, five minus three is equal to minus, is equal to two. Five plus minus three is equal to two. Five plus minus one times three is equal to two, right? Now, how about vectors? Well, let's think about it. What if I will do instead of that, I will do this. Would it be exactly the same? Well, let's just think about this geometrically. Now, this is A, this is B. Now, what's B multiplied by minus one? It's this one, it's minus one times B. Now, this is C which is equal to A minus B. Now, how about A plus, A plus this vector? Well, let's just put this parallel shift to this. This is also B minus one, minus one times B. So if I add A and minus one times B, I will get this. It's very easy to prove, and I will probably prove it in some problem, but it's very easy to prove that these two are exactly the same. They have the same length and the same direction. So this is just an illustration to solution which will be in some other lecture, which will follow this particular. So I was just trying to give you a taste of certain problems which I would probably ask you to solve, and then I'll present the solutions myself. All these problems about these arithmetic operations on vectors to prove that they actually exactly conform to operations with numbers. And one more thing. You remember that vectors have not only geometric representation, but also a tuple representation as an ordered set of numbers. And in particular, set of two numbers represent the vector on the plane, set of three numbers, and a tuple, three tuple, as it's called, represent the vector in three dimensional space. Now, so the question now is how these representations, tuple representations, are changing with arithmetic operations. But this also will be part of the problems which we will be solving together, because after I have defined, I mean, this lecture is just an introduction to arithmetic operation. That's where I define these operations, and how it affects, how these operations affect certain different representation of the vector is a different story, and it will be addressed in subsequent lecture. So far, that's it for today. Thank you very much. I encourage you to go through notes for this lecture on unizord.com. That's in the part which is dedicated to vectors. And I think the topic is called exactly like this, a recent tick, a vector, a recent tick, I think that's how it's called. That's how I call this lecture, vector, a recent tick. So that's it for today. Thank you very much, and good luck.