 Hi, I'm Zor. Welcome to a new Zor education. Continuing the topic of vector arithmetic. In the previous lecture, I was talking about how the addition of the vectors and multiplication of the vectors by a constant can be expressed in a tuple form. So let's just repeat very briefly. If, let's say, we have a vector which has a tuple representation. Well, this is a three-dimensional vector. Actually, it's exactly the same as two-dimensional, etc. Then multiplication by this vector by a constant can be expressed in the tuple representation as multiplication by each coordinate by the same constant. That's one thing which we were discussing in the previous lecture. And the second one about addition, if you have another vector in tuple representation, then addition of these two vectors has a tuple representation when you have add corresponding coordinates to obtain the coordinates of the result. Now, these two fundamental properties of multiplication of the vector by a constant and addition of two vectors are very important in proving a certain properties of the multiplication and addition. Now, the properties which we know from the numbers are commutative law, associative law, distributive law. Remember that commutative law is this one. When a and b are any numbers, the associative law is about additions and about modifications. In addition case, it's this one. You can put the parentheses any way you would like. So first you add a and b and then add c to the result or you add b and c and add this to result of this to a. Same thing with multiplication. You can multiply it this way or you can multiply it this way. And finally, there is a distributive law which is to multiply the number as sum of two other numbers is the same as multiplication of this number by the first, separately by the second and then add them together. So these very nice properties of the multiplication and addition of the numbers can be very easily extended towards vectors because what is a vector, it's just in a tuple representation an ordered set of two, three, or maybe even more numbers depending on the dimensionality of the vector. So if something is true for one number, why can't it be true for a group of two or three numbers, right? And yes, indeed this is true. So this lecture is about all these laws of associative law, commutative law and distributive law. As we know them, we will expand them towards the vectors using the tuple representation. So the group of numbers behaves exactly the same as single numbers basically. That's what we would like to show. Okay, so I presented this particular lecture in the form of problems actually and I'm really asking you to do all these problems yourself because they're very, very easy. If you use the tuple representation to prove all these properties is really a piece of cake. So I'll do it as fast as I can because it's really very, very simple thing. First, we'd like to prove that constant multiplied by a vector can be commutative. Well, very easy. Let's think about this. This is K multiplied by vector, let's say V is V1, V2. I'm using two-dimensional case, three-dimensional, with exactly the same and actually n-dimensional with exactly the same. So K times V is equal to KV1, KV2, right? Multiplication by vector is multiplication of these. Now, what is multiplication of constant by vector we were talking about? What is multiplication of vector by the constant? Well, it's actually from the definition of this, follows that these are exactly the same. Why? Because if you remember multiplication by constant, let's talk about constant equal to a natural number, like 4. What does it mean? You preserve the direction and you increase the lengths by 4 times, right? So multiplication of constant by vector should be defined actually exactly the same as multiplication of the vector by the constant because it has this physical sense. Multiplication means you will just extend the lengths of the vector 4 times or whatever number of times preserving the direction. If K is negative, you remember we were just changing the direction by opposite and it should be defined exactly the same thing. So we can actually say that this is the result of the definition of the vector. It's physical sense. And then obviously from the table representation, it's exactly the same because vk would be defined as v1, k, v2, k. And since numbers, multiplication of numbers is commutative, kv1 is the same as v1, k. kv2 is the same as v2, k because these are just real numbers. Each one of them is a separate real number. So real numbers are commutative as far as the multiplication is concerned. So that's why the commutative law of the multiplication of the vector by a real constant is obvious. Next. Next we will go to associative law, but this is not just the regular associative law. This is a combination of the multiplication of vector by two constants. So we can do it this way, right? Or we can do it this way. So first we multiply two numbers by themselves and then the result by a vector. Or we multiply one number by a vector and then the result is a new vector which is multiplied by another number. Now, why are they equal to each other? For exactly the same reason. So again, if v is v1, v2, then what is this? This is kLv1, kLv2. That's what it is. So the double representation of this is this constant, k times L, multiplied by the first coordinate and by the second coordinate. What is this? Well, these are two different multiplications. First, L times v would be L times v would be Lv1, Lv2, right? Now, whenever we multiply k by this vector, it would be k times L times v1, k times L times v2. Now, is this the same as this? Yes, absolutely, because the multiplication is associated. Same as here. I should really put these parentheses to be more precise, right? Because we first multiply by L. But now kL times v1 is the same as k times L times v1 because the multiplication of the real numbers is associated. Each one of them is a real number. So as you see, the group of numbers behaves exactly the same as single number as far as the associativity is concerned with commutative law. And obviously, we have defined all these operations on vectors exactly having in mind that these laws must be preserved. That's why it was designed this way. All right. So associative multiplication of the vector by two numbers is proven. Next, addition of two vectors is commutative. So v times w equals w times plus v. OK. Same thing, very obvious from the representation. If v is v1, v2, and w is w1, w2, then v plus w is v1 plus w1, v2 plus w2. Now, what is w plus v? Well, this is w1 plus v1, w2 plus v2. But these are equal to each other, right? Because the commutative of the commutative law of addition of real numbers. So if these two top of representations are the same, so vectors are the same, that's the proof of it. As you see, everything is very simple as soon as we switch to a top of representation because the corresponding law of, in this particular case, for instance, commutative law of addition between vectors is reduced to the commutative law of addition of the real numbers, which represent each corresponding dimension. In this case, like two-dimensional can be three-dimensional and a number of dimensions. OK. Next is associative law. Let's say we have a addition. So let's say we have three vectors. So first we add these two, and in this case we add these two. So this is associative law of addition. Now, why are they equal? All right, well, first let's just add these two. Now, this sum in top of representation would be u1 plus v1, u2 plus v2. That's top of representation of this vector. Plus w, which is this one, results would be u1 plus v1 plus w1, u2 plus v2 plus w2. That's this one. OK. Now, let's talk about these guys. First, we summarize these two, which will result in v1 plus w1, v2 plus w2. And now we have to add to this vector, vector u. And u is, I didn't write it down, but u is u1, u2, right? So that would be... So we add u1, which is u1 plus v1 plus w1, and u2 plus v2 plus w2. As you see, we have exactly the same result, which proves the associative law of addition among vectors. Now, let's talk about distributive law. Well, distributive law is when we mix together addition and multiplication, right? So in this particular case, we have a kind of a mix case, because when we multiply, we multiply vector by a constant. So the question is, what are we distributing? We're distributing addition of vectors multiplied by a constant, which is that's the distribution of addition towards multiplication. Or we distribute addition of the constant. So what are we talking about? Well, the answer is both. So let's talk about the first one. Now, v plus w in a table representation would be v1 plus w1 v2 plus w2, in a two-dimensional case. And when we multiply it by k, what do we have? Well, we multiply each constant, each coordinate by this constant, right? So it would be k times this and k times this. This is multiplication of real numbers and addition of real numbers. So the distributive law works. So I can open up these parentheses and I will have this. That's the final result on the left. On the right, I will have, first, kv1 kv2. That's this vector. And I have to add kw1 kw2. And when we add vectors, their coordinates are adding together. So it would be kv1 plus kw1. The second coordinate would be kv2 plus kw2, which is exactly the same as this one. So that's the proof of the first line. Now, let's talk about the second line. Well, when we multiply some, first, you multiply this sum by first component and then by the second component, which we can open up the parentheses because these are all numbers. So it's kv1 plus lv1 kv2 plus lv2. That's our left side of this line. Now, on the right side, well, let's multiply constant k by vector v1 and we will have kv1 kv2 in the table representation. And they have to add lv1 plus, not plus, second coordinate, lv2. And when we add these two vectors in top of the representation, their coordinates are correspondingly added together. So I will have this. That's the first coordinate and this is the second coordinate. So that's my right side. And as you see, it's exactly the same as the left side. That concludes all these very, very simple properties of the vector arithmetic. So just don't forget to add, to multiply whatever vectors is exactly as simply as you operate with numbers as long as you're using the top of representation of the vectors. So it's very easy to add vectors, to just add coordinates. It's very easy to multiply vectors, to just multiply the coordinates by whatever the constant is. We're talking about multiplication by the constant. And whatever the laws of addition and multiplication of real numbers exist in arithmetic, exactly the same laws exist in vector arithmetic. Well, that concludes this lecture. I do suggest you to exercise again these little theorems. They are all presented as notes to the lecture on unison.com. And, well, make sure that you understand all these manipulations with vectors using the top of representation, because that's actually the only representation people are dealing with when they are trying to approach things expressed as vectors analytically. Because analytically it means basically using the top of representation, not the graphics. Graphics are good for some kind of illustration purposes. When you draw a vector and then you add another vector and then this is the sum of two vectors, it's all nice pictures and it all explains quite well. But analytically you have to approach it using the top of representation with coordinates. And that actually implies the dimensionality of the vectors, two-dimensional vectors, three-dimensional vectors, and dimensional vectors, whatever. All right, that's it for today. Thank you very much and good luck.