 Hi, I'm Zor. Welcome to InDesertEducation. Today we'd like to talk about one particular operation on the vectors. It's called ScalarProducts. Yes, ScalarProducts. Well, we operate with different objects in mathematics. And with numbers we can add or multiply them, for instance. With triangles, for instance, we can calculate their area or something like this. So basically vectors are also a subject of manipulation. We know how to multiply a vector by a constant. We know how to add two vectors together. In both coordinate form, the tuple representation, and in geometric form, when basically we know the lengths and the direction of the vectors. Now in the first case, for instance, the direction remains exactly the same, and the length is multiplied by k. If k is positive, the direction is the same. If k is negative, the direction changes to the opposite. Now if you are summing together two vectors, then we're using this rule of parallelogram, which is the result. Now in coordinate form, if you remember, if a is in two-dimensional case represented as a1, a2, and b is represented as b1, b2, they're coordinates, x-coordinate, and y-coordinate of the endpoint of the vector. Then multiplication by a constant means actual multiplication of each component, because that actually gives you the proper lengths without changing the direction. And in case of addition, you just add correspondingly x and y-coordinates. So the sum would be a1 plus b1 as an x-coordinate and a2 plus b2 as y-coordinate. In three-dimensional case, it's this one. Now these are old operations which we have learned. Now this lecture is dedicated to a new operation, completely new operation which we can perform with vectors. Vectors actually came from physical representation of certain objects, like speed, for instance. And Scalar product also has certain physical meaning, which we will address later on. So right now, let's just concentrate on an abstract operation on two vectors. And let's try to examine it first in tuple representation in the coordinate form. And later on, next lecture will be dedicated to their geometric representation. So let's say we have two vectors in their coordinate representation, in their tuple representation. Now I have chosen two-dimensional case. Later on I will explain the three-dimensional, it's exactly the same. So it's just easier to manipulate this two-dimensional case. What is this operation? Well, you know the arguments, two arguments. Now if we add, for instance, two vectors, we get the vector. If we multiply a vector by a constant, we get a vector. If we do this operation which is called Scalar product, or that product, or sometimes inner product, as a result we get the real number. So this operation, if you want, you can use the symbolics of the function. If the function of two vectors is a result, it is some kind of a real number. Now, well, that's fine. I mean we can obviously define many different ways we can derive the real number from these two vectors. For instance, well, we can add their lengths together. Is that okay? Fine, it's a function. You calculate the lengths of the first one, the lengths of the second one, and you add them together, or you multiply them together. Well, this is a different operation. And it has certain physical meaning as we will discuss later on. And that's why I would like to define this operation not as just the formula. I mean, I can give you the formula right now if you want to. What is the symbolic representation of this Scalar or that product? Using the formula, it's this one, in three-dimensional case. In three-dimensional case, I add the third coordinate. I multiply, I'm sorry, my mistake. In three-dimensional case, it will be A3B3, etc. Now, I can give you this formula, and then we can examine the properties of this particular Scalar product, which is a real number, which is a function of these two vectors. But that's not really interesting. I would like to approach this in almost like axiomatic methods. I would like to think about this function as having certain properties, and then derive this formula based on these natural and reasonable properties, which we would like to put into this. Now, why I'm going to do this? Well, primarily because this course is not about giving you some facts about mathematics. It's about to develop your creativity and logical thinking and logic. So, that's why I would like to put some foundation, the logical foundation for this particular operation first, and then derive the formula from these seemingly logical and seemingly reasonable assumptions, almost like axioms of huge. So, what are these assumptions about the function of two vectors which results in some real number? Well, let's think about it. I have actually put something for myself as a guiding point. So, rule number one, I call it independence of coordinates. Now, you see that these are coordinate representation, right? If we change coordinate in some way, in some good way, so it doesn't change the lengths or angles, for instance, we can rotate our coordinate system. From this system, we can go to this system, we just rotate it. Well, obviously, all the coordinates of all the points will be different, and obviously, the vectors also will have different coordinates. However, what will remain in this rotation? Well, the lengths of the vectors, for instance, will remain. And the relative position of two vectors, let's say I have one vector and another vector. No matter how I move the coordinate, the relative position of these vectors, which is actually an angle between them, is the same. So, my point is that whatever calculation with coordinates we make, the result of this calculation should not depend on something like rotation or maybe shift or whatever. So, whatever the transformation is done, which doesn't really change the fundamental properties of the vectors, which is their lengths and their relative direction, which is angle between them, if it's not changed by change of the coordinate, then the result of the scalar product should be the same. So, it's a function of vectors' physical properties like lengths and angle between them, not the result of their concrete representation in a concrete coordinate system. So, independence of the coordinate system is very, very important. It's property of the vectors and their physical essence rather than their concrete representation in some coordinate system. So, that's number one property, independence of the coordinate system. So, the function which operates with these a1, a2, b1 and b2 should not really change the result if we change coordinates in a nice way. Rule number one. Okay, now number two. Rule number two I can express this way. This is zero. Now, what is zero with the bar on the top? Well, it's a null vector. Null vector has zero lengths by definition, right? So, it doesn't have really direction, it has zero lengths. Now, I think it's very reasonable to expect that our scalar product of any vector with the null vector will result in a real number zero. I think it's very reasonable and very logical assumption. Rule number three. This is a unit vector, the vector which has a unit length. Now, I think again it's reasonable to assume that if you have two, if you have the same vector multiplied by itself and the vector has the length equals to zero. So, it's not two different vectors, it's actually vectors which are the same. If we repeat it, we multiply it by itself, then the result must be one. I think it's very reasonable to expect that the vector of the length one multiplied by itself using this scalar product should give the result of this operation should be equal to one. I think it's reasonable. Commutative property. Again, very reasonable property. So, the formula should be symmetrical relative to A and B. If we change their order, the result would be the same. Now, we always expect something like a product, whether it's a scalar product or a product of different numbers to be computative. And also, there is an associative law with multiplication of one of the vectors by a constant. So, if you multiply one of the vectors by a constant and then multiply by another vector using the scalar product, by the way, pay attention, this is a product of a vector by a real number. This is a scalar product of two vectors, this one and this one. I'm using the same dot because all the properties are really exactly the same, like commutative law, multiplication, the associative law, multiplication. And distributive will be as well, that's the next one. So, anyway, this should be equal to K multiplied by a scalar product of A and B. So, this is kind of an associative law, but in this case, we are mixing two different multiplications. This is a scalar product, this is a product of the vector by a constant. This is a scalar product, result E is a real number, right? So, this is a multiplication of two real numbers. Anyway, I'm using the same dot because that's how mathematicians are using these, in these cases, and I'm not going to invent anything else, but you should really understand the difference between this dot and this dot, and this dot, they're all different. This is a scalar product again, this is a multiplication vector by a constant, this is a scalar product again, and this is a multiplication of two real numbers. That's why I put the parenthesis. And, well, obviously, with multiplication of another, it's also the same thing. If I multiply by a constant another component of this, of this scalar product. All right, and the next one is distributive law. Sum of two vectors, this is a vector. Scalar product with the third vector equals to the first component of the sum, scalar product with the third one, then the second component of the sum multiplies, and then add it together. So, this is a true distributive law. So, basically, the laws of scalar product are very natural, as far as I understand. Maybe the law of independence, the first one, might present certain problems, but in general, there is nothing wrong with asking that the product of two vectors depends only on their physical characteristics, like lengths, and direction from one to another, rather than the coordinate system. So, I think it's natural, and as far as these guys are concerned, that anything multiplied by null vector gives zero, and unit vector by itself multiplied will give one. I think it's also quite natural, and it corresponds to your, and mine, understanding of what actually the multiplication is, right? Multiplication by zero should give zero, and multiplication of one by one should give one. Now, so, being as it may, I think it's sufficient number of rules, or axioms, if you wish, to derive the formula for scalar product of two vectors in the coordinate form, and that's what I'm going to do. Now, what I will do first is the following. Think about the vector a, which has components a1 and a2 along the coordinates. I can actually write this vector as the following. As a constant a1 multiplied by a vector 1, 0, plus constant a2 multiplied by a vector 0, 1. Now, y. I think it's quite obvious. If this is the vector, this is your vector a, x component, y component. Now, this piece is a1, this piece is a2. Now, obviously, vector this one equals to sum of this vector plus this vector, because that's how the addition of the vectors is defined. This is the rule of the parallelogram. In this case, it's actually a rectangle. Now, what is this vector? It has a long, it has a length of a1, right? And it's directed exactly the same way as this vector. This is number one. So, the vector from 0 to 1 has coordinates 1, 0, right? Now, the vector from 0 to this point is exactly a1 times longer than the unit vector along the x-axis. So, that's why I specify this vector as a1 times the vector a10. Similarly here, unit vector is here. This is number one. And this vector is a2 times longer than the unit vector along the y-axis, which is 0, 1. So, if I add these two vectors together, I will get my vector a. Now similarly, this is equal to b1 times 1, 0 plus b2 times 0, 1. So, these are vectors in the parenthesis in their top of representation. It's like this one is a top of representation. So, these are unit vectors. 1, 0 is the unit vector along the x-axis. 0, 1 is unit vector along the y-axis. And a1, a2 and b1 and b2 are real numbers. So, this is a true representation of vector a as the sum of two vectors along two coordinates, x and y. Now, let's multiply that. Now, one vector is a sum of two vectors and another vector is a sum of two vectors. So, I can use these laws, very reasonable laws which we have agreed with, of commutative and distributive laws and dissociative laws, whatever, to multiply sum of two vectors by sum of two vectors, right? Now, what will that be? Well, if you multiply these vectors, you basically open the parenthesis in exactly the same way as you open the parenthesis when you multiply sum of two numbers by sum of two numbers. So, it's first times first, first times second, second times first and second times second, right? Because that's how you multiply using the distributive law. So, what if we will do this? All right. Let me put this aside. All these rules I have already specified. I don't need this picture here. So, the multiplication with this follows. A times B equals A1 times vector 1, 0 times B1 times 1, 0. That's first by first. Now, let's again think about what are these dots represent. Well, this is a multiplication of a vector by a constant. This is multiplication of a vector by a constant. And this is this color product. Because the result of this is a vector, the result of this is a vector. Plus, first by second would be something similar. So, the first one is this times this. Now, second by first, that's A2 times 0, 1 times B1 times 1, 0. Plus, A2 times 0, 1 times B2 times 0, 1. All right? Now, using one of the rules which I have already specified, like multiplication of one of the vectors by a constant, actually is the same as you multiply the vector by itself to the second component, and then the result will be a constant. And then I will repeat exactly the same thing the second time. So, instead of this, I can use, well, let me put it on the right, and then I will wipe out. So, it would be A1B1 and then 1, 0 multiplied by 1, 0. That's what it will be. So, I factored out, basically, I took out the A1 outside of the scalar product and then B1 outside of the scalar product. I twice applied the same rule. So, I will replace this with this. Now, similarly, I will replace this with A1B2 and then scalar product of 1, 0 by 0, 1. That would be instead of this component. Instead of this component, I will have A2B1, 0, 1. Scalar product 1, 0. And finally, this one would be replaced with this. Scalar product 0, 1. And this is sum. All of that. So, that's our scalar product represented in this particular form. Now, what is this? Well, it's a unit vector, a unit length of 1 multiplied by itself. And one of the rules is, if you remember, unit vector multiplied by itself should give the result of 1, right? So, instead of this, I can specify this. And that's it. Because this scalar product of 1, 0 by 1, 0 is equal to 1. Same thing with this. This is the unit vector on the y-axis, which is also multiplied by itself as a scalar product will give 1. So, instead of this, I can put this. Now, these guys are a little bit more complex. These are two vectors. They are also unit vectors, but they are not the same unit vector. It's two different unit vectors. And actually, they are perpendicular to each other, right? 1, 0 is here. And 0, 1 is here. 1 is x-axis. Another is along the y-axis. So, they are perpendicular to each other. So, they use this particular rule of unit vectors multiplied by itself equals to 1. Because it's not by itself, it's two different unit vectors. Okay. But let me point out, first of all, that these two are the same because they are commutative, right? 1, 0, 0, 1, 0, 1, 1, 0. So, whatever the result of this multiplication, it's the same in these two cases. So, what is the result of this multiplication? Well, let's just think about what is 1, 0 times 0, 1. Just this particular, what is the scalar product of this? Okay. One of the first rules was independence of the scalar product of the coordinate system. Once, if I take two vectors perpendicular to each other and do the scalar product, the result of this scalar product should not really depend on whether I turn my system of coordinate or I may be reflected relative to some. So, I perform certain changes of the coordinate system without really changing the lengths or the angles, right? Let's consider a different case. Let's consider this case. Minus 1, 1, 0. It's this one. It's 0 minus 1. Now, let's compare this pair, of course, this pair and this pair. Well, in both cases, the lengths of the vectors are 1. In both cases, the angle between them is 90 degrees, the right angle. So, according to my rules of independence of the scalar product of the coordinate system, no matter how you do this scalar product, it should actually be the same. Why? Again, because the lengths of these two is equal to 1 and 1 and the angle between them is 90 degrees. The lengths of these two are 1 and 1 and the angle between them is 90 degrees. So, the scalar product should be equal, right? But now, let's think about it. What is this? I can have this minus 1 as minus 1 as a constant times 1, right? And according to the associative law of multiplication by a constant, I can take this constant out as a constant and then have 0, 1, 1, 0. So, this is a scalar product and this is multiplication by a constant. And what do we see? Look at this. This and this. This is the same, right? So, we have something like x equals to minus x, right? Where x is the scalar product of this. So, what is x in this case? Obviously, x is equal to 0, right? That's the solution to this equation. So, my point is that multiplication of two unit vectors, the scalar multiplication, the scalar product of two vectors of the unit length perpendicular to each other, the result is 0. So, there are no other components in this sum. They are all equal to 0 and only the first and the fourth one remain. I have derived the formula like this. I'm not giving you as a definition of the scalar product in the coordinate form because, again, you might actually think about why. Why is this particular formula not something like, I know, a1, b1, a2, b2? This is also a real function of all four coordinates, right? But it's not this one. It's this one, this particular expression, which is actually an expression which satisfies all these reasonable, as I'm considering them, rules which I have specified in the very beginning. And one of the very important rule is an independence of the coordinate system. So, what I want to say right now is that if I will change the coordinate system, I will, let's say, rotate it. I will change it in a way which does not change the length and the angle between two vectors. If I rotate the coordinate system, the vectors basically have the same length and the same angle between them. Then, formula would be different, but the result would be the same. Coordinates will be different, but the result will be the same. That's very, very important to understand. Okay. Now, obviously, if I will change the coordinate system in a way which is like stretching, for instance, when the length of the vectors is changing, then all bets are off, obviously. This is not a transformation of the coordinate system which we are talking about. We are talking only about the transformation of the coordinate system, which does not change the metrics of the space, does not change the length, the angles. These, by the way, are called orthogonal transformations of the coordinate system. Rotation is one of them. Symmetrical reflection relative to, let's say, y-axis will be exactly the same. Some others. All right. That's what I wanted to present to you today. This formula, which again is not a definition, I derived this formula using some logic. Well, basically, we have created a new operation called Stuller Product based on certain reasonable principles which we put in the beginning. And that's very important because how mathematicians actually came up to this formula. They didn't come up just from the blue. They were thinking about we need a formula which would reflect certain properties which we would like to preserve, like, for instance, independence system transformation, distributive law, associative law, multiplication by number, et cetera, et cetera. And then they logically derived this particular formula. So that's what I would like to, like, inculcate in your minds. All these formulas are not just given to us by God or something like this. We have come up with these formulas as a result of logical conclusions based on certain principles which we consider reasonable. We can actually take them as axioms. Okay. So this is a representation in the coordinate form. And again, let me point out if I will change the coordinate system in an orthogonal form so it doesn't change the matrix, then obviously the coordinates will be different of these two vectors, right? But the result of this formula calculation would be exactly the same number. So the scalar product is invariant, so to speak. It does not change with relative to orthogonal transformation of coordinates. Okay. That's it for today. I do suggest you to go through notes to this lecture on unizor.com. And the next lecture will be about geometrical representation of the scalar product. Thanks very much and good luck.