 Hello, welcome to this lecture of bio mathematics. We have been discussing about vectors, trying to understand quantities that have both magnitude and direction and we discussed a few things about force etcetera. So, for example, in the last lecture, we have been discussing how the charges will move, you know, in which direction if you have a set of charges and if you ask the question, what is the force on a particular charge due to all of the charges, which is a common question that would be needed to understand shape of biomolecules, because biomolecules have charges and they have charged molecules, charged residues and charged molecules. So, essentially, an effective shape of a biomolecules will be determined also by its charge-charge interaction. So, in this context, we have been discussing how will charges feel a force due to other charges and which direction, which will be the direction of the force etcetera. Today, we will continue to discuss a little more about vectors, but from a different point of view. So, today's lecture is also vectors and it will be vectors, the third session. And what we had discussed is about basically force as we saw, this is our, where we stopped last time, that we said the resulting force on the second charge is a sum of the force due to the second on the due to the f 2 1 and f 2 3. f 2 1 is a force on the second due to the first charge and force on the second due to the third charge. We also, and we said that this is f 2 1 and this is f 2 3. So, we also said that essentially, we will get f 2 is equal to some alpha, which is along x and with some minus beta, which is along y. So, the force, if you look at the x y plane, this is something along x axis, which is alpha, which is positive. So, you have to move along this direction, some amount of alpha and minus beta in this direction, along the minus y direction, alpha along the plus x direction in this direction and beta along minus y direction. So, this will be the resultant direction of the force. So, this will be the direction of the force of this due to this charge 2 and we have charge 1 and charge 3 here. So, this is the kind of thing we found out. Now, we will discuss a few, we also, so to some, while we learning all this, we learn many concepts. Those concepts are like, those concepts are here, which are finding out resultant force, idea of unit vector, how do we find the magnitude of vector, how do we find direction of a vector. So, all these things we learnt. We also briefly discussed about addition of vectors, how we can add two vectors. So, let us generalize this to three dimension. So, we know that any vector, when three dimension will have three components, just like a vector in 2D has x and y component, a vector in 3D will have three components x, y and z. So, any vector a in three dimension will have three component, some component on the x axis, some component along the y axis and some component along the z axis. So, if you take the generalized 3D vectors, how do we add them and subtract them? So, we have a look at here. So, if we have such vector a and b, where a is a 1 x plus a 2 y plus a 3 z and b is b 1 x plus b 2 y plus b 3 z, the sum of this vectors is nothing but the sum of the components. This is what we discussed. That is, if you can just a 1 plus b 1, which are the x components, a 2 plus b 2, they are the y components and a 3 plus b 3, they are the z components. So, these are, this is how you find the sum of two vectors, a plus b is a 1 plus b 1 x plus a 2 plus b 2 along y hat plus a 3 plus b 3 z hat, a 1 plus b 1 x hat plus a 2 plus b 2 y hat plus a 3 plus b 3 z hat. So, this is the resultant. This is the sum of two vectors. This is how we find. So, like we said, if you want to find out f 2 1 plus f 2 3, we can find out this way. Similarly, subtraction, addition is, subtraction is like addition only. So, if we have two vectors a and b and if you want to subtract one from the other, you can do it in the following way. If you have a vector a, it is a 1 x plus a 2 y plus a 3 z and b is b 1 x plus b 2 y plus b 3 z, where all this, when I say x y z, they are cap the unit vectors a 1 x unit vector plus a 2 y unit vector plus a 3 z unit vector. So, then, z is equal to a minus b, which is, you subtract the x component, there is a 1 minus b 1 plus a 3 a 2 minus b 2, which are y components and a 3 minus b 3, which is z component. So, this is the way we add or subtract two vectors and they will have, the resultant vector typically will have, it can, it can have three components or if you have vector of 3 D, you add subtracted in 3 D, you will get another vector in 3 D. That is what you saw here basically, because what you get the c is another vector and the vector is obtained in this particular way. c is a, c is another vector and the c is obtained in this way. Even here in the previous case, when you do a plus b, c is another vector. So, you can do this in a particular, in this particular way. Now, so, we learned about subtraction of two vectors and addition of two vectors. Now, we should go and rethink a bit more about something we discussed very early about vectors. So, when I said that if we have a 2 D plane x axis and y axis, any, so this is x and this is y, any point in this x y plane, let us say this particular point, if you want to go to this and let us say, you can imagine for example, these are atoms of a molecule. So, like we can imagine some molecule, which are connected like this, let us say these are 3 atoms of a molecule, this is one atom, this is one atom, this is one atom. So, if you want to, if I want to tell this position of this particular atom, I can say that if I start from this particular point and if I go, so I want the position of this point can be represented by some vector r. Now, what is r? We said that you can go some distance along x axis and though you will get the, and you can go some distance along x axis and some distance along y axis. So, if you go this much along x axis and this much along y axis, in the direction of the y axis, I will get this particular point. So, r is some x along the x direction plus some y along the y direction. So, this is your r vector. Now, if you think about it, to get this exactly the same information, you can do, you can get the same information in a different way. So, have a look at here, instead of saying that I can go x along this, x distance along this and y distance along this. So, this is one and this is this. I can say one thing. I can go an angle theta, some particular angle from the x axis and if I go this angle theta, then I can go a distance of r, this distance. So, let this is the distance from here to here, in this, in this direction. So, I can specify the same point by specifying this distance and this angle. So, I can specify some distance r, which is this distance. So, this is not a vector, it is just a distance. So, let me call this L. So, L is the mode of the r vector. So, this is L, this is this distance from here to here, this is the distance, let us call it L. So, I can specify this L and specify this angle theta. So, that will give me, that will give me the exact position. So, somebody, if I just specify L alone, that will not tell you this particular point, because if I say this is a distance, my point or this particular actum is at a distance L from the center, at this, from the starting point. There can be many points, this point also can be, there can be many points in the, on the circle, you can draw a circle here. There can be many point, that is L distance away from this, even this point is L distance away from the center, even this point is L distance away from the center. So, there can be many points, actually along a circle, which is L distance away from the center. But if we specify this L and this angle theta, where this is angle from the x axis, then there is only one unique point, that is this point. So, this point or this position of this atom or this particular position can be either represented by an x and a y, such that r is x e x plus y e y. So, let me write this little more carefully here. So, let me write this. So, let us have a look at little more carefully. So, we have x axis and y axis and any point can be specified by specifying this distance. So, let me call this distance L and this distance see this angle theta. So, you can either specify an L and a theta or you can specify x and a y. If you specify in x and y, this r is x e x plus, sorry, r is x x cap plus y y cap. So, x and y are some numbers and x cap and y cap are unit vectors. Same way, I can specify with a length and a theta. So, sometime I represent this L by r. So, or some people write it as r and a theta. The distance r, some people write L instead of r instead of l. Some in many textbooks, you would see r and theta. So, either you can, essentially r or l, whatever you represent, it is this distance. So, either you can say this distance and say that you can go this distance at an angle theta from the x axis or you can say that I can go x along the x axis and y along the y axis. So, you will get this. So, now, if this is the formula for this r vector, what is the formula in terms of r and little r and theta? So, that is the question. So, let me say, let me, let me, let me, let me explain what I was selling in the last two minutes. So, what I was saying is that any point, look at here, any point that is the end of this blue line, this blue arrow, this point can be represented by numbers 3 and 4 in 2 D. You can have two numbers. In 3 D, you will have three numbers. So, two numbers, three and 4 will tell you this, uniquely tell you this point. When I say 3, 4, in a 2 D plane, there is only one point, 3 along x axis and 4 along y axis. When I say 23, 8, there is only two points, 23 along x axis and 8 along y axis. So, two numbers will uniquely determine one point in a 2 D case, in a 2 dimension, in a plane. Now, as we said, how do we say, instead of saying two numbers, I can say two distances, essentially these are two distances, distance along x axis and distance along y axis. Instead of saying two distances, I can say one angle and a distance. So, how do we do that? As I just said, I can say this angle theta and this distance r. If I say this way, it turns out that x, that is this distance, is nothing but r cos theta and this distance y is nothing but r sin theta. So, if I say, in this particular language that I was mentioning here, so if you look at this particular case here, this is our point of interest and we have, this is L is this distance. So, here this is the x. So, and you know that cos theta is this distance, which is the y distance, this distance. So, cos theta is the definition of the cos theta is. So, if you consider this as a rectangle triangle with this as a 90 degree, this is its hypotenuse. So, cos theta is always x by L, that is the adjacent side, which is adjacent to this angle theta, which is x divided by this hypotenuse L. So, this is the definition of cos theta. So, this would imply that x is L cos theta. In this case, you can convince that x is L cos theta. Similarly, you can convince yourself that x y is similarly, if you do this, you will get here that sin theta, sin theta is nothing but, you can say that sin theta is nothing but y by L. So, in other words, this implies that y is L sin theta. So, here you have L cos theta and L sin theta, where L is the distance. Just like I say in this slide here, this R, which is the distance is essentially x is equal to R cos theta and y is equal to R sin theta. So, this is just simple trigonometry. We will tell you that this distance is only, you know that cos theta is always between, for any, it is always between 0 and 1. It cannot be more than 1. It is either 0 or any value, which is some value between 0 and 1, including 0 and 1. It cannot be more than 1. So, if this angle theta, if there is an angle theta, this distance is always less than this distance. You also always know that if you have a rectangle triangle, the hypotenuse is R, this is basically x square plus y square. So, let us think about this. So, R, so if you have, let us draw this again, x and y axis and you have a vector R. You have a distance R and this is theta and this is your x distance and this is your y distance, sorry, this is your x distance and this is your y distance and this is, this are the x axis and y axis. So, this is your x distance and this is your y distance and we just said that, we just said that x equal to r sin theta, sorry r cos theta. So, x is equal to r cos theta and y is equal to r sin theta. Now, let us check this using some trigonometric idea that we know. So, if we know, if we have any rectangle triangle, so this is a rectangle triangle. So, this is the 90 degree, right, this is 90 degree. So, this is a rectangle triangle, this line, this line and this line, this, this three forms a rectangle triangle and we know the famous Pythagoras theorem, which says that the r square, which is this, is x square plus y square. r square is x square plus y square. So, now, let us do this x square plus y square here. So, if you do this x is r cos theta and y is r sin theta, if you do x square plus y square, which is r square cos square theta plus r square sin square theta. So, r square is common. So, r square into cos square theta plus sin square theta and we know that cos square theta plus sin square theta is 1. So, this is r square. So, essentially, what we knew from the Pythagoras theorem is essentially here. So, this is a simple trigonometric idea that we used to get this x square plus y square is equal to r square and that is essentially, that is what x, this means x is equal to r cos theta, x is equal to r sin theta only means some trigonometric identity. But, this has an interesting implication that x component, that is what this x is, this x component is r cos theta and y component is r sin theta. So, this has interesting consequences like we were discussing charges. So, if we had one charge here and two charges here and if we say that the force is in this particular direction and if we know that particular angle. So, let us say we know this particular angle. So, let us say that you know this angle or you know this whole angle. If we know this particular angle, we can calculate the x component and y component in the manner that we specified. So, this is interesting. So, one exercise will be for you to check the x component and y component you got for the force in the previous day. Can they be represented as a cos theta and a sin theta? So, that is something which you would want to check. So, when I say the vector f 2 as we yesterday's force, this can be written as some kind of x cos theta plus y sin theta, where x and y. So, x sorry, this can be written in terms of cos theta and sin theta. So, f 2 can be written in the cos theta. So, we after today's lecture, you would be able to write this in a position. You will be in a position to write this force in terms of cos theta and sin theta. So, we have taken the first step. We have understood how x component and y component can be written in terms of cos theta and sin theta and how does this r square now depends on r square is essentially how do we calculate r square that is something that we want to check. So, now before check, so what if something is r square, r square is nothing but r into r, r times r is r square. So, to understand that you would need a bit about to understand about product of two vectors. So, that is what the next idea that you learn that product of two vectors. So, the question we get we want to ask is how do we find the product of two vectors. Now, you know that vector has a magnitude and a direction and we also show that if you add two vectors or if you subtract two vectors, you always get another vector. Then the question is if we find the product of two vectors, will we get a vector or not. So, this is some question we can ask. If you find the product of two vectors, will we end up with another vector or a scanner. So, let us think about it. So, let us think about have a look at here. So, we have a pen here and if I apply a force on this pen, the pen moves. So, force is a vector, force is a vector. The position of this pen is a vector. You can always represent an x axis and a y axis and the position of this, position of this can be represented by a vector. This particular vector represents the position of this particular the top of this pen. Now, I can apply a force to change this position. I change the position from here to here. So, I introduced a displacement. So, I change the position from here to I change the position. Now, the new position is this. So, essentially the way this this this top this particular thing moved by certain distance. So, this is a displacement. So, this distance D is a vector. It moved in a particular direction, something which was here previously, moved from here to here. So, when this is moved, I I applied a force. I applied a force F such that it moved in this particular direction. So, you have a force which is a vector. You have a displacement which is a vector. Now, the product of force and displacement we know is energy. So, what is energy? Energy is a is it a vector or is it a scalar? It a as you know energy is a scalar. Energy has no particular direction. So, since energy is a scalar, it is clear or it it occurs to us that the product can be a scalar. So, a product between two vectors can be a scalar. So, it turns out the reality is that you can have the product of two vectors can sometime be a scalar, sometime be a vector. So, you have two situations, two possibilities. The product can be either it can be either a vector or a scalar. So, have a look at here. So, what I is you can imagine two situations. Product of two vectors is the scalar that is one situation that is we just discussed. Example, when you apply a force and change the position of an object. For example, the product of a force and displacement is work and the energies the work is a scalar or the energy is a scalar. So, the product of force and displacement is an example for a something called a scalar product, where the product is a scalar. You can also know what is the example that you have where you have the product is a vector. So, again if I look at here, I can apply a force and then move this in a particular way in a. So, you know something can rotate in a particular way. So, let us say this way of rotating. This is we can say what we are generating is this is it is a counter clockwise rotation. Something is rotating in a counter clockwise. I can also apply a force, I can apply something and rotate it in a clockwise direction. So, what you have to apply to do this rotation is called a torque. You have to apply a torque. So, torque is some quantity which is a vector, which is again the product of the force you apply and this distance. So, the product of this force and this particular distance, if I try to rotate it here or try to rotate it apply a force here or if I apply a force here, the torque is different. I can rotate I can rotate it more by applying a force here. So, the torque is essentially a product of force into the distance from this particular pivot from from where it is rotating. So, anytime like all of you might have seen, when you try to open a door the handle the the force you apply on the handle is making it to rotate. So, the essentially what you are applying is a torque. So, the torque is again force to rotate something. So, the torque is essentially a vector, which is also a product of two vectors. We will see that. Another example which is biologically very. So, the torque which when we opening the door the torque is something that we see everyday life. . Another example which is biologically relevant will be one example will be let us say if you want to twist a DNA. So, DNA you know that they proteins twist the DNA and they get supercoiled and so on and so forth. So, if you want to twist a DNA you have to apply again a torque. So, proteins apply a torque on DNA. So, the torque has a particular direction. So, this is some quantity which is product of two vectors, but is again as a vector. So, two vectors and the product is a vector is a torque. Again in bacteria in bacterial flagella you know that the flagella flagella or motor can either rotate clockwise or counter clockwise. This is another example where things you have to apply a torque which is again product of two vectors. You can also holding something right. If you want to hold something this muscles will have to apply some force apply some torque. Otherwise if there is a heavy object which will try to fall under gravity and if you want to hold them this muscles or if you want to if you do this if you want to push pull this up essentially you are applying a torque. So, there is a particular direction for this. So, all this are some vectors we will we will we will explain this little more carefully, but what I am trying to say is that there are objects which are product of two vectors, but they have directions. Such objects are the one example of such object is torque which is very you are very familiar when you opening a door or even in the biology when you when you talk about supercoiling there is some direction associated with it and the you do apply some torque to supercoil the DNA and such torque has direction. So, essentially we will discuss them, but then what we want to understand is how do we find this product of two vectors. If we are given two vectors how exactly will you find the direction of these two vectors. So, that is the question that we are we are interested to ask. So, let us say the product of two vectors the second point I wanted to make is that the product of two vectors another vector and that is example is applying a force to twist DNA applying a force to rotate an object. So, these are examples where you have force and some displacement essentially distance essentially giving to a torque or a twist. So, now how do we find the Scala product? The Scala product is defined in this particular way. So, let us say you have a vector f which is some x component a 1, some y component a 2, some z component a 3 and some distance x which is some x component b 1, some other x component b 2, some other x component some x component b 1, some y component b 2, some z component b 3. Then the product the Scala product which is written as f dot x. So, this is sometime is also known as dot product, because this is represented typically by a dot in between this and this is nothing but the magnitude of f times the magnitude of x times the cos of the theta between f and x. We have two vectors you can always define an angle between these two vectors. So, this f into x into cos theta. So, this is what it is. So, let us have a look at carefully how exactly this is defined. So, we said that I have some object here. So, we had discussed this. So, this is some objects here. If I apply a force in this particular direction. So, my force is in this particular direction. Let us say I am applying a force in this particular direction and it is moving in the same direction and it is moving in the same direction. So, the angle between the force and the distance, we have force and we have a distance. So, let me explain this again here. You have something here an object here and you want to apply a force in this particular direction to move it. So, this is your direction of the force. So, let me apply this force in this particular direction. So, applied this force and made it to move. So, I made it to move in this particular direction. So, this is the direction in which it moved. So, this is the d. So, the force, which is the d, they are in the same direction. So, the angle between the force and the, so you have vector force and the d, which is this displacement vector. So, the angle between f and d is 0. So, the theta in this case is 0, no angle. They are going in the same direction. Then, the f dot d. So, let me, let us write f dot d. This is the product. This is mod of f, mod of d into cos of theta, which is angle. Here, the theta is 0. So, cos theta is 1 when theta is 0. So, cos theta will become. So, this will be just mod f, mod d. So, this is the typical energy that we talk about. f into d, when we say, we are talking about, you are applying a force and it is moving in the same direction as you apply the force. You need not move. If I apply a force in this direction, it can move in some other direction. I can apply a force in this direction and the object can move in this particular direction. So, then there is an angle between this. So, then the work done is mod f, mod d cos theta. In the last class, we discuss how do we find the mod of a vector. So, if we have any vector f, which is a 1 along x axis and a 2 along y axis and a 3 along z axis, the modulus of this, we said, is root of a 1 square plus a 2 square plus a 3 square. This is what mod of this. We also said that, similarly, if you have a vector d, which is d 1 x plus d 2 y plus d 2 z, then f dot d is mod f, mod dot d, mod d, if they are in the same direction. So, this is basically root of a 1 square plus a 2 square plus a 3 square into root of d 1 square plus d 2 square plus d 3 square. This is how you find out the energy. So, this is basically the energy. This is the scalar product. Now, if there is an angle, you have to take the angle also into account. So, you will get f d cos theta. We just… So, now, let us take a simple example. Let us take an example f dot f itself. So, we… Let us say f is some vector. Let us call this f 1 along x axis plus f 2 along y axis plus f 3 along z axis. Now, let us say f dot f. What is f dot f? This is like finding square. So, this is mod f, again mod f. And the cos of angle between… If you have same vectors, there is no angle because f is the same direction as f. There is f and f. They are the same thing. So, the direction is the same. So, cos 0 is 1. So, it is just f times f. So, now, what is mod f? Mod f is root of f 1 square plus f 2 square plus f 3 square into same thing f 1 square plus f 2 square plus f 3 square. So, this is square root. Square root goes away and you will get f 1 square plus f 2 square plus f 3 square. So, f dot f is f 1 square plus f 2 square plus f 3 square. So, this is an interesting result that will be useful if square of a vector is f 1 square plus f 2 square plus… This is how you find the square of a vector. So, when you say f square or any square of a vector, you will have this particular form. So, you know, this is nothing but mod f square. Mod f into mod f. So, you know, mod f is square root of f 1 square plus f 2 square plus square. And square of this is this. Now, let us have a look at another one. So, let us, again, go back to two-dimension, so that everything is simple. So, let us think of a vector in 2 D. So, you have a vector in 2 D, which is this vector. And this has some x component and some y component. So, this is the x component and this is the y component. Now, we already learnt that this can be represented as x is equal to r cos theta and y is equal to r sin theta. We already learnt this, where this is the angle theta. Now, this is the unit vector in this direction. So, let us say a small vector, unit vector in this direction can be represented as x cap. x cap is a unit vector in this direction. And let us call this vector r. So, x is this distance. This distance is modulus of r, r vector. Modulus of r vector is this particular distance. So, you know that this distance times cos theta and this distance times sin theta is x and y. .. Now, let me also, we can also define r as x component along x axis and y component along y along y axis. So, or let me write this r 1 along x axis and r 2 along y axis. This is r 1. So, x, you can write r 1 if you want. And this distance, this distance y, you can call it r 2. Now, if we find r vector dot x cap, what is this? What is r 1 r vector dot x cap? It turns out that if you do this, according to our definition, this is mod r vector mod x cap vector into cos of this angle between theta is angle between this x and this r. Now, mod of x is x cap is 1. So, this is r cap r times the modulus of the r vector times cos theta. So, have a look at this. So, r dot x cap r vector dot x cap is nothing but mod r cos theta, which is the same thing that we described here. So, this r cos theta is nothing but mod r dot x x. If you do this, you will get r cos theta. Similarly, just viewing this, you can also see that r vector dot y cap is mod r sin theta. You can use simple trigonometry that you already know and find out the dot product between the r vector and the y f, unit vector along y. So, this is all related to each other. So, this is one thing that you should know about the dot product and another kind of thing that there are few things that you can deduce from this, which we will come to, we will come, we will discuss this later. A few things that we can use the way you can use the dot product in biology. Of course, calculating energy is a big example, because force and displacements are very common. You might have heard of something called molecular dynamic simulations. Molecular dynamic simulations are something that is extensively used in biology to understand protein folding or dynamics of proteins or dynamics of biomolecules. So, when we set up, when somebody sets up molecular dynamic simulation, they have to get exact idea about vectors, positions and the energy. So, they have to calculate the force in the molecular dynamic simulation, what they are essentially doing is solving a set of differential equations to get the positions. So, they have to use the idea of all this idea of calculating the force between the charges that we discussed. They have to calculate the position of each atoms. So, they are represented by vectors, they will have x component, y component and z component. So, the position of each atoms are represented by vectors and the forces, there are each charge or each atom or each amino acid, each atom in an amino acid will be feeling force from due to all other atoms. So, basically you have to calculate the resultant force due to all other atoms to do this molecular dynamic simulations. So, we have to calculate the resultant force that we discussed, you have to calculate the position vectors and then you have to calculate the energy. So, energy is nothing but the force times d, f dot d just we discussed. So, you have to also use actually the idea of a Scala product. So, everything that we learnt so far that is the idea of vectors. So, finding the resultant force and position vectors and the dot product to calculate energy etcetera has to be used if you want to really do a molecular dynamic simulation for example, or if you want to do theoretically any calculation of protein folding or any calculation of dynamics of biomolecules if you want to do those calculations you basically have to understand whatever we have been discussing so far. Now, what about the next one which is vector product? So, what is the definition of a vector product? So, the definition of a vector product is basically if you have a vector r which is r 1 x plus r 2 y plus r 2 z and f is f 1 x plus f 2 y plus f 3 z, you can define the torque t as r cross f. So, this is called cross product. This is because the r and f are vectors and the resulting torque is also a vector. Why is the resulting torque a vector? Because now what is the answer? So, this is basically mode r mode of sine theta this is angle of theta between this and n is a unit vector which is representing the direction along which the torque will act. So, what is the direction of the n that we will discuss later, but n will represent is a unit vector that represent the direction of this torque. So, the essentially the answer r cross f is basically r cross f is basically a vector and this kind of products are known as scalar products. Sorry, this kind of products are known as vector products where the resultant the result is essentially a vector r cross f is a vector. So, you can think of simple thing that we do every day that is again go back to opening and closing the door. You can apply if you apply in a particular direction the force you will open the door if you apply the force in another direction you will close the door. So, for this particular torque there is a plus or minus sign if you wish. So, or in other words if you look at here if you look at if you look at this particular case of applying this is the distance r. So, I can call this as the distance r. So, let me let me keep this here. So, this distance let me call this r this is my r vector. So, this vector is my r vector. Then what I am doing is I am applying a force here. So, if I apply a force in this direction the torque. So, this will rotate in this particular direction if I apply a force in this direction it will rotate in this direction. So, this will be counter clockwise rotation and this will be a clockwise rotation. So, there is a direction of the force this is if you call this plus f and if you call this minus f this is a force you are applying in the y direction. So, let me call this y direction. So, f is some f 0 along the y direction here I am applying the force which is along the minus y direction. So, minus f 0 y. So, I can either represent along plus y direction or minus y direction. So, depending on that the torque which is defined as t here is basically the r which is always in the same direction cross f. So, f sometime in the that direction or the other direction. So, depending on that the direction of the torque will change. So, we said that r f sin theta. So, we know that the theta which is this angle this angle is 90 degree here. So, when 90 degree theta is 0 and see sin theta is 1. So, it is essentially r f and there is a unit vector n. So, the answer is r f n the modulus of r modulus of f n now what is n. So, it turns out that n will be always in the direction perpendicular to the direction of the force and the and the r. If f and r are in this 2 D plane the n will be in this perpendicular direction. So, this is one way where you want to know about this scalar product. So, we will we vector product and we will come back to this and tell you this in little more detail whenever we need to use it for dealing with some examples, but for the moment you just understand that this is the case. And one more idea which I want to say today briefly and which we will we will elaborate in the coming classes is something called gradient. So, how I look at here that is briefly if imagine this is concentration of some protein and the concentration of the protein is more here and this less here it is dark here and this is not so dark here. So, the concentration is changing in this direction, but if you look at here in this direction is not changing in this x direction the concentration is changing. So, to represent that this direction for the concentration is changing we can introduce something called gradient. So, the gradient is some vector called del. So, concentration is a scalar. So, gradient of a concentration can be represented by t derivative of c with respect to x with an x cap. So, this is some definition which we will come back later and try and help you understand in the coming classes, but so far today we learned a few things and we will stop with this today. So, to summarize what we learned today we learned about plane polar coordinates. So, there is an error in what I wrote here. So, it is essentially plane we talked talked about something called plane polar coordinates. This is basically have a look at the slide and the plane polar coordinates r and theta you can represent any vector by a distance and an angle theta. So, that is called plane polar coordinates and we also learned about dot product and cross product two way of finding the products of vectors. We also learned about the gradient we briefly mentioned about the gradient and we will discuss this in the coming one more class in the coming class about vectors. We will discuss about the gradient and how will that lead to diffusion equation etcetera. So, with this I will stop today's class. These are the ideas that we learned and we will continue the discussion in the next lecture. Bye.