 Hi, I'm Zor. Welcome to a new Zor education. We are continuing talking about vectors and operations and vectors. So far, we have learned about additional vectors which produce the vector, multiplication of a vector by constant, which is also produced as a vector. Then we were introduced to an operation which produce a scour. So we have two vectors and an operation which is called a scour product or dot product, that produced just a number, real number. And obviously we investigated certain properties of this operation. Now we will talk about another operation which is also named product. So there was a scour product and the result of the scour product is a scour. Now we will talk about vector product and the result of the vector product is a vector. So two vectors are engaged in this operation which is called vector product or cross product. And the result is a vector. There is one very important distinction. Other operations like addition, multiplication by number or scour product, were actually defined for one-dimensional, two-dimensional, three-dimensional vectors and actually can be expanded to any n-dimensional vectors. Vector product is a specific operation which is defined only on three-dimensional vectors by definition. Now why such an interesting situation? Well, primarily because the vector product is an operation which was invented for purposes of physics. Now we live in a three-dimensional world and the operation on vectors in three-dimensional space where the vectors represent certain physical characteristics of our space. That was actually the fundamental reason for introducing this operation of vector product. And I will just very briefly mention one of the physical characteristics of this particular operation. If you can imagine a magnet which has north and south poles, so there are magnetic forces which are in between these two poles. Now you can feel these magnetic forces if you will put some ferromagnetic, like a piece of iron for instance. It will gravitate towards one of these two poles. Now here is another interesting story. If you have an electric current going this way somehow, let's say you have a metal rod and you have an electricity which is moving. Let's say we were connecting this to some kind of a battery. So there is an electricity which is circulating along this metal rod. So we have a metal rod where electricity is going on and then we have the forces of magnetic origin. And they are perpendicular to each other. Then the experiment shows the certain force which acts on this rod. And it's perpendicular to the surface. So if everything is going on in the surface of this white board, then the force is perpendicular to the white board. So this metal rod would actually act upon towards either that way or this way, depending on the direction of the magnetic forces and the electricity. If you change the polarity, this force will change the sign. So if it was pushing that way, then changing polarity will result in pushing it this way. If you change the poles, north and south, also the direction will be changed. So we have this situation when you have two different vectors. One vector representing the electric current, another vector representing the magnetic current. Then there is a third vector, the force, which results in some interaction between these things. So that was the reason actually why Vector product was invented. I mean this and some other physical experiments which were done. There was some kind of a necessity for mathematicians to basically introduce this abstract. Now we're talking about abstract operation, which is called vector product. But you have to understand that the roots of that were in physics. Now we will talk about the vector product as actually a mathematical object. However, we will always keep in mind that there are some physical foundations, which this mathematical apparatus should actually represent. All right, now let's think about. We are talking about two vectors acting in certain plane. And the result is the vector which is perpendicular to this plane, right? So that's the basic of that physical experiment. So when we were talking about the vector product, we're talking about two three-dimensional vectors. And operation of vector product produces a third three-dimensional vector. Now to define this vector, we have to define its magnitude and its direction, right? Okay, so magnitude and direction. Let's think about this experiment again. If I will change the position of my metal rod where electricity is going on from perpendicular to the magnetic lines to a parallel ones, the experiment shows that there will be no force generated. And actually the force depends on this angle. So if magnetic force is going this way and electricity going this way, now if I will start changing the direction of the rod turning it a little bit, then the force will be diminishing more and more down to zero when instead of this direction I will have parallel to magnetic field. So what I would like to say that this vector product is somehow depends on collinearity of these two vectors. So the more collinear they are, the less magnitude of the result you should expect. By the way, this is opposite to a scalar product. Because in a scalar product case, the collinear vectors have the biggest result, the largest result if you scholarly multiply them. Why? Because it's magnitude of one times magnitude of another times cosine of an angle between them and the cosine in this case is zero. So the cosine, the angle is zero, so cosine is one. So when they are collinear, the magnitude of this result is the biggest. In this case, case of the vector product is just opposite. So when they are collinear, the result is zero. When they are perpendicular, the result is maximum. That's what experiments show. So instead of using the cosine for scalar multiplication to basically measure the collinearity, we will use the sign. And that's what's very natural to use. So what I would like to say is that the magnitude of the vector product of these two things actually is naturally vector product. It is naturally dependent on sign between their directions. And obviously it depends on their magnitudes themselves. So that's, I should say, this. The magnitude of their vector product is equal to the product of their magnitudes and the absolute value of sign of the angle between them. So this is the definition of the magnitude of this vector. Now, let's talk about direction. Well, first of all, I did tell you that the force is always perpendicular to this plane. So I can always say that no matter what vectors A and B are, the result of their vector product is perpendicular to them both. So if you have one vector A and let's say another vector B, now let's imagine that I'm drawing everything in a three-dimensional. So the result will be perpendicular to both, considering that they are, A and B are in this plane, so to speak. And A cross B is perpendicular to the plane, which means it's perpendicular to this and it's perpendicular to this, to both. So if you have two vectors, no matter how you position, you can always consider a plane which they define. Two vectors always define the plane. Now their vector product is perpendicular to this plane. So the line is defined absolutely without any problems. Now the magnitude I also defined. So the only thing which is not defined is direction, this way or that way along this line. So again, we have two vectors. They define the plane. Now this is perpendicular to the plane, which means it's perpendicular to this and perpendicular to this and perpendicular to any other vector which is on this plane. Now the magnitude is defined, so it's only the direction which we have to define and we will talk about this in a second. Okay, let me wipe out this. Don't need this anymore. And let's talk about direction. So this is one vector A, this is one vector B, and this is A cross B. Now this is an angle. Now here is the rule. And again, this is a rule which is a definition. There is no like some kind of physical sense in it. It's just a definition. We have to define somehow the direction of this vector. If we know the line it's within and we know it's magnitude, so it's only the direction, like upwards or downwards or whatever you want to call these two directions. Okay, here is the rule. If you have to move from vector A to vector B within this particular plane which these two vectors define, so if you have, if this is your A and this is your B, this is the plane, so the direction from A to B. And we're always talking about the smaller angle, not this one, this smaller angle. So if you have two vectors, we always can talk about the plane they belong to and the angle from which we can move from one vector, which is the first one in this case, A. This is A to the second one, which is B. So we move this way. Now this is the rule which is called right-hand rule. Imagine you are opening a bottle of wine, you have this corkscrew and you are moving it this way and it goes down, right? So let's imagine that you have to move from this A vector to B angle in the same direction you are moving the screw. So if you're moving the corkscrew, for instance, you have to move it in this particular direction and your direction of the tip of the corkscrew this way, right? So that's the direction which is, by definition, is the direction of their vector product. So if this is A and this is B, then this is the direction of their vector product. So from A to B, if you move to a certain direction, then you can imagine you do it with a corkscrew and that's the direction where the tip of the corkscrew is moving, right? Another way of viewing this thing is if you have a plane, let's say this is a plane, it doesn't really matter, which defines these two vectors. Now this is the line which is perpendicular. So you can view this plane from underneath or from above, right? So which side to choose? Well, you have to choose the side from which the direction of the angle is counterclockwise. So if you look from this, from the below, it will be counterclockwise, which is the right one because if you move from here to here, you go downwards. Now if you look from the upper side above this plane, if this plane is considered to be horizontal and this is above, now you see that the movement from here to here from A to B is clockwise, which is a negative direction if you remember when you are changing the angle. The positive direction is always counterclockwise. So this is not the side of the plane where the vector is directed. It's always towards the side from which the direction is counterclockwise, the positive direction. This is the rule. So you can talk about the right hand screw or cork screw or you can talk about the direction of the movement from underneath or from above the plane which contains these two vectors. Whatever the definition is, it's always defined and it's always the same. So in this particular case, if you consider the angle between A and B as basically from direction from A to B is this way, then my proper direction of the product should actually be down. If, however, this is A and this is B, then the direction is this way from A to B and then this would be the direction of the cross product. So we know the magnitude of the product, of the cross product of two vectors and we know the line along which this result, this vector product is supposed to be positioned. It's perpendicular to both vectors. And we also know both are right angle. And also we know how to choose the direction on this line. The direction depends on the right hand rule. Basically that's what defines the vector product. So the vector product is completely defined by all these characteristics. It's one three-dimensional vector multiplied by another three-dimensional vector. The result is yet another three-dimensional vector. That's why it's called vector product. Now it's notation little cross. That's why it's called sometimes cross product. And we know how to basically find the result of this operation, this vector product. Now let's talk about properties. So the vector product obviously has certain properties. We learned about scalar product and its properties. Now we're learning about vector product and its properties. Alright, let's just go one by one. The first property. If you multiply any vector by a null vector, vector product would be, well, a vector, right? But it's a null vector, which has zero lengths. Why? Because the magnitude of this vector is the product of the magnitudes and the sign between them. Now the magnitude of zero, null vector is zero. So that's why the magnitude of this vector is zero, so it's a null vector. Which doesn't really have any direction, so it doesn't really matter to talk about. What's interesting is the next one. Let's think about this. Well, the magnitude of the spine is magnitude of this times magnitude of this. Lengths of this vector times lengths of this vector. But what's the sign of the angle between them? Well, the angle between them is zero because it's the same vector. So the sign is zero. The result is also a vector which has a zero length, which is a null vector. This is a little bit more unusual, I would say. Because if you remember, for instance, in case of a scalar product, you multiply these two things in the scalar product mode and you basically have a double length. Because the cosine of an angle of zero is one. So that's the difference, that's a very interesting property. So the result of the vector product of a vector by itself is equal to zero. And it corresponds to the physical experiment which we were talking about. Because if I put my, these are magnetic forces. If I put my electric rod, a metal rod with electric current parallel, there will be no force. That's what the experiment actually shows. So if one vector is parallel to another vector, their vector product, which actually is representing the force, which results in this experiment, is zero. But as soon as I start turning this, it will actually be growing. And the formula, again, it was probably experimentally proven. The formula would be that the force depends on both strengths of magnetic force and the electric current and the sign of the angle between these directions, between the directions of the magnetic current and the directions of the electric current. And if it's zero, well, then the result is zero. But if it's 90 degrees, like here, if my conductor is this way, this metal rod, then the strengths of that force which acts in this direction, perpendicular to this plane, is the maximum. So that's an interesting property which is very peculiar for vector product. And again, it corresponds to physical characteristics. Now, what's another important property is? We all know about commutative law, right? Like this is, Scala product is commutative. Why? Because it's A times B times cosine, and cosine is an even function. It doesn't depend on the direction of the angle. But now this actually is vector, the length of this thing is equal to lengths of this times lengths of this times sine. So the sine is an odd function. It changes the direction, it changes the sine if it changes the direction of the angle. So this is called anti-commutative law. Now the anti-commutative law is very important in this particular case. Why is it anti-commutative if you will just look at this particular picture? Well, if you move from A to B, then your right hand rule says you go this way, right? So that's the direction. If you move from B to A, now B is the first vector and A is the second vector. You have to move this way and it goes down. So the magnitude would be the same, but the direction of the vector would be opposite. That's why this minus stands here. Alright, next. Next property is very interesting as well. Let me just draw a little different picture. Now my A and B vectors will lie in the plane of the white board, right? This is A, this is B. Now their vector product would be lying along this line. Now, if you move from A to B, then this is counterclockwise. Therefore, if my observer sits here on this side of the white board, it's positive direction. So the vector actually directed this way from the white board towards me. Okay, now what is, now this is phi, absolutely. Well, let's just think about it. This is the length of this. This is the length of that vector. Now what is, let's say, B times sine of phi? Well, obviously this is an altitude of this parallelogram, right? So if you build a parallelogram on these two vectors by basically taking the end point of one vector and drawing the line parallel to another and sensing here from the end point parallel to another, you get a parallelogram. And the area of this parallelogram is equal to base times altitude. And the altitude is exactly B times sine of phi and the base is A. So the vector product is a vector but its length of this vector is equal to the area of the parallelogram which is built on these two initial vectors. So that's just an interesting property. Which we will use in another three-dimensional problem later on. Okay, now what else is interesting? Well, next is quite obvious associative towards multiplication by a constant. So if you have a vector A and B, you multiply A by a constant. It's a scalar. It's just a number. And then you cross multiply by vector B. It's the same thing as if this constant is multiplied by vector product of A and B. And similarly, if the constant is multiplied by B, it will be exactly the same thing. Now how can that be proven? I mean, before all these little properties I proved quite easily. Well, actually this is not difficult as well. Let's just consider for a second that K is positive. Which means it doesn't change the direction of the A. Well, if it doesn't change the direction, it changes only the length. Now, length of this vector is the length of this times the length of this times sin of this. Well, absolutely. Now, the length of this guy is absolute value of K times the length of this guy which is this and absolute value of sin. Now, the length of the K A is exactly the same as absolute value of K times A. So that's why the length is exactly the same. The length is the same. We can find out this. Now, the line along which the product actually is would also be exactly the same. Why? Because multiplication by a positive constant doesn't really change the direction. So K A, if this is A, this is B, then this is K A. So the direction is exactly the same. So the plane which is defined by these two vectors A and B, or K A and B, exactly the same. So the perpendicular towards this plane line is also the same. So the magnitude is the same. The line along which the vector product is the same. So only the direction we actually talk about. But again, if K is positive, K doesn't really change the direction of the A and the direction of the angle from A to B is exactly the same as from K A to B. So for positive K, it's obviously the same direction. Now, for negative K, we don't need this anymore, right? Now, what if K is negative? But here's what happens here. We change the direction of this to the opposite, right? Which means instead of this, K A would be here for negative K, right? Now, if angle used to be this, now angle is this one. We are talking about a smaller angle from the beginning, from the first vector towards the second. Which is changing the direction, obviously. If this is phi, this is 180 minus phi. The sine of phi is exactly the same as sine of 180 minus phi. But the direction is different. And since the direction is different, it doesn't change the magnitude because the sine is exactly the same. But it does change the sine. So this one is changing to the negative. That's why this one is changing to negative. And that's exactly what K does. K is negative right now, right? So whatever used to be the sine of A times B, now it changes to the opposite, and same thing in this case. So that's a little bit longer probe, but it's still an easy one. Now let's talk about another property, a very interesting property. Imagine that we have a Cartesian system of coordinates, or Euclidean system of coordinates. Euclidean space, Cartesian system of coordinates, whatever. Now we have unit vectors, which have lengths one, and directed along the three x's. So i is a unit vector along x, j is a unit vector along y, and K is a unit vector along z. Now what's interesting about these angles? Well, number one, they all have lengths of one, because these are unit vectors by definition. That's number one. Number two, they are mutually perpendicular to each other. So i is perpendicular to j, j is perpendicular to K, and K is perpendicular to i, right? It's three-dimensional space. Let me use mine. These are unit vectors. i, j, K. So they're all of lengths one, and they all are perpendicular to each other. Now, let's talk about vector product. Let's say i times j. Well, the angle is 90 degrees. So the sine is equal to one, right? The magnitude, therefore, is equal to one. Magnitude one times magnitude of another times sine of the angle. And direction of the angle, look from i to j, it's from here to here, right? So if we are talking about a quark screw, we go this way, which means it goes up. So we have basically this particular vector, K, which is exactly positioned on the perpendicular to the whole plane where i and j are. It has the lengths one, which these are supposed to produce as a vector product. And direction is right. It's upwards, right? So what I want to say is that i times j equals K. Isn't that interesting? So vector product is producing a result if you multiply i times j, these two unit vectors, you will get K. Now, very similarly, if you multiply j by K, you get i. Go back to my three markers. This is j. This is K. So you go from j to K, which means from here you are doing this kind of a turn, and obviously this is a direction. And it's perpendicular to these two. So that's what this is doing. And finally, K times i use j. K times i use j. Now, obviously if you change the order, instead of i times j, you have j times i, you will get minus K. Right? So because the vector product is anti-commutative. Anti-commutative, right? So these are properties of unit vectors along the axis of coordinates in Euclidean space. That's very interesting property. And the final property, which I would like to talk about, but very briefly, is distributive law. You obviously expect the distributive law. So if you have a vector product of some of two vectors and the third one, then it's the same if you add the vector product of the first one by the third and the second one by the third. Well, this is a true property. To prove it is not so trivial, and I will delay it until the next lecture. However, as a property, I would like to mention it in this lecture and this will be the very the very end basically of the properties which I was going to talk. This is the last property I wanted to discuss today. So the distributive law is working as well. Well, now the vector product, I would like to mention again that the vector product is actually the result of some physical experimentation and development of the physics. Mathematicians basically took it under their wing and started to research, investigate, prove theorems, et cetera, et cetera, so it grew into a mathematical topic. However, again, do not forget that the basis is over there. Now, is it right to call it vector product and why this particular operation? Well, yes, obviously it has certain physical meaning and we have to research it. But at the same time, there is nothing which prohibits us to invent a completely different vector product. I don't want to actually call it vector product. Product of two vectors. Let's put it as multiplication of two vectors. Let's say you have vector in coordinate system, one and another. Now, there's some, if you remember, was this. Some of two vectors is a vector where you sum the coordinates and it does make actual physical sense as well when you have two forces, for instance, and you have the result of this. Now, why don't we define multiplication of two vectors as this, as a vector with two coordinates? I'm not putting plus in between. The plus means it's a scalar product. I'm actually putting as a comma here. So it's two coordinates. I just multiply them. I mean, why can't we do this? Well, we can, quite frankly. There is nothing wrong with this and this operation would probably satisfy lots of very useful principles like multiplication by zero, by new vector would be, new vector multiplication by one, one, for instance, if this is one, one, then the result will be the same as before. Commutative law, that's the works. Associative law works. By the way, associative law is not working. If you multiply by third vector cross by c, there is no associativity there. But in this case, you will have, so it kind of defines a nice operation. But for whatever reason, it was not needed for any practical purposes and mathematicians basically ignored it. I mean, yes, we can do something like this, but nobody's interested. But many people are interested in the vector product because it has a true application in physics. So I don't want you to think about vector product as something strange, unusual, or whatever. I always want you to think about vector product as a mathematical reflection of something which exists in the physical world. Now, with this thought in mind, let me finish this particular lecture. I do suggest you to read notes for this lecture again. It's like a textbook, basically. It's allinunisor.com. And next lecture will be actually about this distributive law and non-trivial proof in this particular presentation kind of thing. All right, that's it for today. Thanks very much and good luck.