noc19-ma19-lec05

So, today we are going to start our last lecture of the fifth lecture of the first week. And here we are going to introduce, what is called a cross product of two vector. That is I will define a new kind of multiplication of two vectors, and I will call this as a c vector it is a new. So, this product of two vector does not give me a number just like the dot product it gives me a vector. So, let us see what do we mean by this, and let us from practical perspective also we will see what happens. So, here is the x y z plane sorry space. So, if a is a one vector here, and say b is another vector in space. Now, c is the cross product of two vector is a vector, which is if you write a cross b, and c vector when you write a cross b you move from a towards b. And then in the plane formed by the vectors a and b, the c vector or the dot product or this are the cross product is perpendicular to the plane in which the vectors a and b are lying. So, why did I draw c in this direction and not the other direction. So, it is like you know moving screw, if you take a screw and put it in front of your nose straight. And if you try to rotate it in the counter clockwise direction, counter clockwise from your point of view. Now, if I am rotating it in front of me, it is counter clockwise from my point of view, but it is clockwise from your point of view. So, from my point of view if I move it in a counter clockwise direction, you will see the nose of the or the tip of the nail tip of the nail or the screw will just point towards my nose. So, that is exactly what is happening. So, if I from here a to b when I am moving it from a to b, this the tip is moving towards this direction. So, I am just going to demonstrate to you through kind of experiment here. So, it is kind of here like this and suppose I am moving it like this, you see what happen is that. So, this is how a screw is moved. So, this is a screw and I am using a screw driver, I am moving it from suppose there are 2 vectors which were defining the plane in which this face of the screw is lying and I am moving it like this. Then I am actually putting the screw, I am moving it like this. So, you see that it is the bottle is going if I it is a it is clockwise. So, this is a screw which you have got and you are moving it. So, this is a screw that you want to put in and you are moving in this direction clockwise from your direction and see what is happening the screw is getting inside. So, basically clockwise. So, if I was standing there then for me it would appear that I was putting it counter clockwise and it was going inside like this. So, in this case may be I am little mistaken I think I have to this case it might be if I am moving from a to b, then I would see from my side it is coming to the counter is going on the clockwise direction. So, it C should be something like this. So, how do you define this vector? So, there is a way of defining this vector. So, we will have a way. So, we will take a vector as a 1 I vector where a 1, a 2, a 3 are coordinates of a. So, we are just in three dimension now because beyond three dimension you cannot define the cross product, but three dimensional ideas would actually be helpful in the sciences in natural sciences like physics and you will very soon see their applications at the towards the end of the course. When we talk about Green's theorem and Gauss divergence theorem, Stokes theorem see how important this concept of A cross B is and a large amount of electromagnetic theory which combines the forces of electricity and magnetism into one framework which Maxwell did also needs this kind of operation. So, this is what it is and B vector. So, the coordinates of b 1, b 2, b 3. So, I can represent it as b 1 I vector, b 2 j vector plus b 3 k vector. Those who know linear algebra you know this is nothing but expansion through the standard basis which is I j and k which is 1 0 0 0 0 1 and so 1 0 0 0 1 0 and 0 0 1. So, I am not getting into all these details. You know these things very well pretty well we have been writing this for quite some time now. Now, let me tell you how do I define it. Now, I how do I define it how what is A cross B. This is the very strange kind of definition where you will see that I am talking about a determinant. I am talking about a kind of determinant. I want you to compute the determinant where one of the rows consists of this I j k vectors I j k. You might say how can you do it because determinant you always put in some numbers. How will you suddenly telling oh my god this has to be something like this. This has to be kind of this is a metric and how many these are all vectors how can you compute the determinant. Of course, I can say you can write down this whole thing in terms of the three vectors and then do the determinant. Then you will also get the same answer, but we will not get in the detail. So, how can I write this? So, what I am writing what are the components of A and B with the I vector A 1 B 1 here A 2 B 2 with the j vector and with the k vector. So, this is the definition. So, you might ask how this definition is meaningful because I j k are vectors and not numbers, but I can say I can write down them as numbers as 1 0 0 0 1 0 and 0 0 1. Then I would like to write the expand it in the form of determinant. Then also you will get back the same kind of stuff. So, let me just expand the determinant and let me just see how can we write it. So, what did we have here is that two vectors A cross B. So, computing determinant is a same it is not that I am doing something it will it will look like exactly the same. So, it will be the same thing here, but I will write it in a slightly different way. So, this will be a real number you take the determinant of this which will be scalar multiply would make a scalar multiplication with this vector. So, it will be A 1 A 3 because if I take I and I take of this first row and first column. So, sorry A 2 A 3 B 2 B 3 this multiplied by I vector. Now, minus sign would come just like in ordinary determinant. Now, I will have A 1 A 3 B 1 B 3 plus this which is multiplied with the j vector and A 1 A 2 and B 1 B 2 which is multiplied with the k vector. So, that is what is how you define the cross product. And what let us now look into again as I showed you through this that the determinant of A is that if this is A and this is B and if I am rotating from A to B in a clockwise fashion the C vector is in this direction. And if I rotate it in a anticlockwise fashion the C vector as I told you just before with the screw I am showing this example if I am rotating in a anticlockwise fashion anticlockwise towards B and you see that this nose of the bottle is coming towards B. So, this is this is what it does there is a very important kind of stuff that I really you really need to know before you do anything with this we need to put certain properties of this vectors into perspective. So, let us put some properties. So, number one property is that if I have A cross B these two this is the negative of B cross A because the directions will change because here it is a vector it has both magnitude and direction. So, we have not computed the magnitude of A cross B what is the magnitude of A cross B we will soon do that. And you have suppose now I will distribute it is like it follows a kind of similar laws of arithmetic that if I distribute this cross product over the addition of two vectors that is A cross gamma B plus mu these are real numbers of C vectors scalar multiply. So, this is nothing but gamma of A cross B plus mu of sorry not gamma gamma is real number A cross C. So, this is this is what you will have this is a property. So, distribution of multiplication over addition. So, again distribution kind of the reverse kind of thing means basically you have multiplied from this side you can multiply from the other side also. So, I would not do it suppose you have gamma A vector plus mu B vector and this you have multiplied with C vector you can easily write down what will be the answer I leave this to you. But what you get from this is the following that if you know this property then let us look at the first property what does it say if I put B is equal to A. So, if for if I put B is equal to the A vector it tells me immediately that A cross A vector is same as the negative of A cross A vector which similarly shows that twice of A cross A vector is 0 which immediately again shows. So, this is simple any vector which is equal to its negative must be the 0 vector it cannot be otherwise. So, this is a interesting property once you have known this property you can play around a bit with i and j. So, let us write down what you will have now let us look at the three dimensional picture again with i j vectors in tau do not mind my drawing. So, this is x axis y axis and z axis and this is the i vector with coordinate 0 0 1 it is a 1 0 0 and this is a j vector with coordinate 0 1 0 and this is a k vector with coordinate 0 0 1. Now, one thing is clear from the very beginning that i cross i is equal to j cross j is equal to k cross k that must be 0 from what we had just discussed. Now, what is i cross j that is if I move i towards i from j. So, basically if I am looking from this side i and j is in the x y plane if I am looking from this side and I am moving i to j. So, from my side I am looking at it. So, I am if if if the observer is standing along the z axis is i is along the z axis then he observes that the i moves towards j in a counter clockwise fashion. So, that the screw will move towards him along the z axis that is if is i is on the z axis it is the k vector which will move towards him. So, i cross j is the k vector similarly, if I have j cross k. So, again if my i is on the x axis and if I move the vector from j to k if I rotate from j to j towards k side then I am rotating the whole system in an anti clockwise direction. So, the i vector will move towards me. So, I will have j cross k as i vector and I will have j cross k as k cross i as j vector this is a beautiful simple setup I think this is something you should always remember because this can be used in many many many places. Now, I will talk about something called a triple product a triple product and we will see what a triple product is a triple product of 2 vectors is that you take the dot cross product of 2 vectors and take the dot with the third vector. So, that is called the triple product this triple product is a very useful one as you will see very soon triple product. So, you have 3 vectors given to you a b and c and where a vector as before is given as a 1 i vector a 2 j vector and a 3 k vector while b is given as a 3 j vector and c is b is again given in the similar fashion I think which I should not repeat b is given as b 1 i vector, but still I am doing it for the sake of clarity I guess and c is c 1 i vector c 2 j vector and c 3 k vector. So, triple product is this the triple product the triple product is defined as follows triple product is a cross b intersections is a dot product with c. So, I take the cross product of the 2 vector and take a dot product with c. Now, if I expand it. So, what I will get I will get the following. So, I will get triple product is a 2 a 3 b 2 b 3 i vector minus a 1 a 3 b 1 b 3 j vector plus a 1 a 2 b 1 b 2 as k vector and this dot product into c 1 i vector c 2 j vector and c 3 k vector b 1 b 3 j vector plus a 1 a 2 b 1 b 2 as k vector and this dot product into c 1 i vector c 2 j vector and c 3 k vector if you recall the definition of the dot product which we had already given. So, then this is nothing but. So, this whole triple product I am writing it here. So, that you can see it clearly. So, this is nothing but c 1 into c 1 into the determinant of a 1 a 3 b 1 b 3 minus c 2 into the determinant of a 1 a 2 b 1 sorry a 1 a 3 b 1 b 3 plus c 3 into the determinant of a 1 a 2 b 1 b 2. So, it can be written either as a determinant whose first row is c 1 c 2 c 1 c 2 c 3 just this has to be correction a sorry it is a this is a c 1. So, it is a 2 a 3 and b 2 b 3. So, if you look at the. So, what I did the dot product the first component of this first coordinate of this with the first coordinate of this and second and second coordinate with second coordinate third coordinate with third coordinate and sum up. Now, if you look at this thing this can be either written as a determinant with the first rows as c 1 c 2 c 3 or it can be written as a determinant as a third row as c 1 c 2 c 3, because the sign convention of the first row and third row are the same. So, actually the determinant if you just just try to remind you that is the determinant if I if I have 9 positions. So, you can start expanding from any of the rows. So, then if you go by the rows in plus minus plus minus plus minus plus minus plus minus plus this is the way the convention is carried out. So, just keep it in keep this in mind and. So, if you do that for our convenience just for writing convenience I am writing it as a determinant which is expanded with c 1 c 2 c 3 in the third row sorry a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3. So, what do you get? So, what I get is that naturally because a dot b a cross b is a vector and c is a vector. So, the dot product of 2 vectors is a scalar, but this scalar can be expressed as a determinant with the inputs which are the coordinates of the vectors a 1 and a 2 and a 3 and this will become pretty important as we go along this this concept. Now, I want to find the area I want to find the magnitude the length. So, my question is what is the length of a cross b magnitude what is the length or the magnitude just slight length which is much colloquially easier length of a cross b in some sense I am asking to determine the norm of a cross b and. So, what is the norm of a cross b that is essentially that is exactly what I am trying to ask you. Now, how do I do it? I have to first note that a cross b is nothing but a vector whose coordinates are this this and this which are determinant. So, if I write it as a square norm of the norm square norm square is then square of the value of this determinant square of the value of this determinant square of the value of this determinant which means if I go back and write it down the first determinant is a 2 b 3 minus b 2 a 3 whole square plus a 1 b 3 minus b 1 a 3 whole square plus a 1 b 2 b 1 a 2 a 1 b 2 minus b 1 a 2 whole square. Now, this is the length the square of the length and now I will write it in slightly a modified way I will just I will square up these as make a square and then rearrange the terms and then I will get the following expression which I am just writing it down I I ask you to check it. So, from here to here your job is to take a cup of tea and sit down and do the check. So, the checking if that will give me a 1 square a 2 square a 3 square into b 1 square plus b 2 square plus b 3 square minus a 1 b 1 plus a 2 b 2 plus a 3 b 3 square. So, this is a very quite cumbersome computation and just I mean lazy and just because of the time constant I am leaving out this computation and asking you to check this computation. And then what happens if I go back please understand that a is a vector with a 1 a 2 a 3 as coordinates and b is a vector with b 1 b 2 b 3 as coordinates and you can understand this term a 1 b 1 a 2 b 2 a 3 b 3 is just the dot product. So, I can write this whole thing as norm a square norm b square minus a dot b of course, I should have the arrows on the top whole square. So, this can be written as norm a square. So, if I have the two vectors a and b and theta be the angle between them and I can write this as this square minus norm a norm b cosine of theta where theta is angle between two of them. So, this will give me that norm a cross b whole square is norm a square into norm b square minus norm a square into norm b square. Cosine square of theta and as a result of this I can write norm of a square into norm of b square which we can take out from the expression from the bracket this is again the distributive law multiplication over addition standard distributive law of arithmetic. So, every time we are actually doing this sort of computation which looks very common and very obvious to us we are actually doing a very deep operation. We are actually writing down certain kind of laws of arithmetic which was initially written down because they were figured out to be true with numbers with natural numbers and then it was written down as a law that a number which if it has to a real number it has to follow all these things. So, they do follow. So, this is not just on the top of it looks like it just an obvious operation, but it is not. So, very simple as we think this is a kind of axiom kind of thing which actually happens. So, we put it as a kind of law which will we always have to remember and follow that this is nothing but distribution of multiplication over addition the kind of laws I had written down there. So, this is the thing if you just think that these are real numbers and this is just multiplication and this is just addition that is what that is exactly what it is here. So, those who know trigonometry would immediately realize that sin square theta plus cos square theta is 1 and. So, as a result of which sorry I have to put a and b here the arrows I always seem to forget the arrows no. So, I will have here sin square of theta. So, noting that length is always positive I simply have this equal to a b norm of a norm of b sin of theta. There is a deeper geometric meaning of this and that is what I am going to work out here on the board that what does this thing actually means, what does cross product mean geometrically. And you will be surprised to know that the geometrically the cross product simply means the area and that is what we are going to do will show that the area of the rectangle contained by the or the parallelogram sorry not rectangle parallelogram contained by the vectors a and b is nothing but the cross product a cross b. And that is what we are going to demonstrate, but I am looking for a duster. So, the duster here let me try to demonstrate this some time is usually lost when I am rubbing the board, but that is how life is that is how things are done in the class. I sit here before an imaginary audience and I am speaking to you as if I am giving a lecture before you in reality. So, let me now write down what that stuff is. So, suppose in the three dimensional space here is my a this is the origin right and here is b maybe I should make it. So, here is your may be the angle I should make it it might immediately in the space it might look as if I have drawn something perpendicular I may be actually not I should write it like this. So, this is my a vector this is the origin and this is the b vector. So, let us make out a parallelogram out of it. So, may be something. So, it is parallel to this and something which is parallel to this. So, this is the parallelogram. Now, what I am trying to claim is that a cross b represents the norm of a cross b that you see here actually represents the area of the parallelogram contained made by these vectors a cross b in the plane that contains the vector a cross b because how do you compute the area of a parallelogram in a plane. So, here is the area of the parallelogram see this is one. So, what do you do area of a parallelogram how it is computed how does one compute the area of a parallelogram. So, this is the height. So, you basically compute this side and suppose this is as a length L and this has length suppose this has a length b and this has length L and this is theta this line the height h is b sin theta and the parallel area of the parallelogram if you define the area. So, what is the area suppose I have constructed this rectangle like this and because they are between the same parallel lines the area is same this is what you learn in your school geometry, but though it is not. So, simple we can we are doing things very simply, but it is not. So, simple. So, this h equal to b sin theta is a height basically if you take the height basically this part and this part because this is also theta and this part. So, that would. So, this height and this whole part this part would make up which is same as this. So, this length and this length is same. So, this whole thing is actually L. So, if you look at the area of the rectangle that is form area of the rectangle is L h b sorry L L into area of the rectangle is L into h. So, rectangular area rectangle rect is L into h length into the breadth, but that is what is L into what is h it is b sin theta. So, you observe that this length this line this side of the rectangle must match this side of the rectangle, but this line is same as L because this is a parallelogram opposite sides are equal. So, this length is also L this full length and. So, it is L into h is area of the rectangle and the area of the rectangle is equal to then L h into b sin theta and this area of the rectangle is same as the area of the parallelogram. So, if you consider this part length as b vector and this is theta and this part the length is a vector of this and you know then what is the answer. So, the area of the area of the parallelogram this parallelogram this parallelogram this one this this one. So, I am trying to find the area of this area of the parallelogram is nothing, but norm a norm b the lengths into the angle between them sin of theta and what is it it is nothing, but the norm of a cross b. So, a cross b this vector in some way represents the area. So, the vector a cross b could be either in this direction or it could be in the this direction perpendicular to the plane it it depends on which way you are rotating the a and b a towards b or b towards a, but this vector say a cross b or this one whatever you want to say whichever is a cross b this vector is actually representing the area and hence area itself can be thought of as a vector the area vector and this would be very helpful when we try to study Stokes theorem and Gauss theorem which are extremely important in engineering and physics. So, please understand that cross product gives a the whole goal of cross product it is to tell you that we can view the notion of area as a vector we have always thought of area as a scale or a number, but area can also be viewed as a vector. So, area vector of an object is actually perpendicular to the object whose area you are trying to calculate length for example, the the plane for example, the rectangular. So, the cross product of two vectors is actually the area of an a rectangle. So, area of a rectangle can be also viewed as a kind as a vector and that is that is where the beauty lies in this whole thing. So, let us now do two things before we finish our talk is let us look at the geometry of 2 by 2 determinants means basically I am trying to look at what is this I am trying to look at determinant of this form. Now, we will soon see something pretty interesting. Now, suppose this is a plane x y plane and you have two vectors a with which is a 1 and with coordinates a 1 a 2 and another is a vector b with coordinates b 1 b 2 and this is the angle theta between them. In a three dimensional setup their coordinates are a 1 a 2 0 b 1 b 2 0. Now, if I look at the cross product of these two vectors a cross b then the cross product is nothing but I sorry there is no comma here j and k a 1 a 2 0 b 1 b 2 0. So, this is nothing but the determinant of a 1 a 2 b 1 b 2 k vector. So, the parallelogram in the plane the magnitude of the parallelogram what is the magnitude of this vector. So, magnitude of a cross b in this particular case is nothing but you take the magnitude on both side norm of the k is 1 right this is this is the k vector. So, if you take the magnitude of this vector and so now here here we are looking at a cross b as if it is a three dimensional vector. So, I am putting a 1 a 2 0 and then what I am getting the magnitude of this is nothing but a 1 b 2 minus a 2 b 1. So, this determinant what does it give it gives me the area of the rectangle area of the parallelogram formed by the vectors a and b in the two dimensional plane x y. So, two dimensional determinant the determinant of with two entries is actually representing the area of a parallelogram. Similarly, I will show the determinant of with three when I when I am talking about a determinant with three entry when a nine entries basically three entries in row each row three dimensional determinant. Then that determinant actually would then speak about the volume of a parallelo pipette because the the notion of a rectangle the notion of a rectangle. So, a notion of a parallelo gram not rectangle notion of a parallelo gram in three dimension becomes what is called a parallelo pipette and that is what I am going to show that a determine the area or the value of the area of a parallelo pipette with which is made up of three sides by the vectors a b and c are actually is nothing but the determinant of over form this. Which is the cross product of three vectors. So, suppose I draw a parallelo pipette I so I am three dimension. So, this is my origin. So, let me see I will which way I will go drawing a vector b vectors. So, a vector b vector and this is my c vector. So, the parallelo pipette you can form like this I am not I am trying to draw this, but something like this will be something like this. So, this parallelo pipette a this is called a parallelo pipette. So, area of the parallelo pipette is actually. So, how does. So, we are showing how we can we geometrically interpret the two notions that we are developed today. So, let us have a. So, here is your a cross b suppose this is your a vector and this is your b vector and this was your c vector. So, a cross b has some kind of is a perpendicular to the plane assuming that I am moving like this a cross b this is my motion. So, a now what is the norm of what is what is if I take the dot product with c what is that. So, a cross b into c is nothing, but norm a. So, this angle is some psi. So, norm a sorry it is norm of a plus b it is norm of a plus b into norm of c into cosine of psi. So, norm that is what you have here that is that is how that is the definition of the dot product. But again what is c cos phi what is this c cos phi giving me this c cos phi is actually this it is this height this is actually the height if I replace this parallelo pipette with a rectangular parallelo pipette standing between the same two parallel bases. So, c c this is the height. So, again if I take the base the area of the base rectangle and multiply it by the height then I get the area of the area of the rectangular parallelo pipette. So, basically and that rectangular parallelo pipette has should have bi-euclidic geometry the same though it is 3 dimensional you can just take your vision further you can just take a leap in the imagination and know that the rectangular parallelo pipette which is standing between the same two bases and this parallelo pipette must have the same area. So, this thing is nothing but the area of the parallelo pipette because this is this is nothing but the height. So, the base the what is the area of the rectangular parallelo pipette the base area into the not area of the parallelo pipette I am making a mistake sorry I should be correct it is volume. So, I am 3 dimensional there is no area talking about. So, volume of the parallelo pipette please forgive me it should be volume of the parallelo pipette. So, the volume of the parallelo pipette is that this is nothing but the base the base area into the height. So, if you know how to calculate the height of a rectangular parallelo pipette is what the base area into height. So, but that rectangular parallelo pipette has the same area as this parallelo pipette the this slanting parallelo pipette. So, volume of the parallelo pipette. So, this is nothing but the volume of the parallelo pipette this is the volume of the parallelo pipette and so, what is happening what is the volume on that parallelo pipette I know what is a cross a dot a cross b dot c it is a 1 b 1 c sorry a 1 a 2 a 3 b 1 b 2 b 3 and c 1 c 2 c 3 which I already know. So, because the triple product is nothing but this determinant. So, this 3 dimensional determinant is nothing but it represents the volume of parallelo pipette 2 dimensional determinant represents the volume of a sorry area of a parallelogram. So, here geometry both 3 and 2 dimensions are linked to the notion of determinant. So, algebra and geometry are linked and that is exactly what all mathematics is all about that algebra is somehow geometry and geometry is algebra. Once you are get a feeling about this then you are you will start enjoying mathematics. So, I call it off for today and we move forward for the next days one which will come along it will be we will be going now towards higher dimension. So, we will no longer be in 3 dimension we will be in 3 2 you know 4 5 6 up to n dimension and we will show by some examples why this n dimensions matter why you have to go beyond 3 dimension. One comes from physics from special relativity another comes at very good example comes from linear programming and that is what where we are going to show that we really need to talk about dimensions more than 3. We really need to talk about something called R n where n could be 5, 10, 20 anything even a million right. So, thank you for your patience and if there is any problem you can always write it down on the portal of the course and please go back and listen to this lecture once again that the core idea is that algebraic definitions and manipulations are actually inspired from geometry math is all about visualization. Thank you very much.