 Now, before we start this chapter, I would like you to understand what all we are covering under this. Basically, it is actually called by many books as vector algebra. Vectors is a concept which you have already done in physics before. So, we are going to learn what all algebra is applicable to them. Algebra itself means addition, subtraction, multiplication, any kind of an equation formation that can happen. All those things will be covered under this concept of vector algebra. Okay. So, in vector algebra, we are basically going to talk about number one, the concept of first of all, introduction, which will be very fast because you know, what is vectors, it's just a formality for us to know everything. Yeah, yeah, why not? Why not? We can have a matrices of vectors. Yes, we can. Okay. Okay. Then we'll be talking about operations on vectors. Now, this operation will include a lot of concepts like addition, okay, subtraction. When it comes to product, there will be a slight detailing that is going to happen here. We'll be talking about product of two vectors and under that we are going to still study two types of product of two vectors. One is called the dot product, which probably you are using in physics to calculate the work done by a force. Then we are going to talk about cross product or vector product that is used to study the moment of a force torque and all are cross f is calculated by this. And we are also going to talk about product of three vectors, which is probably removed from your syllabus. NPS, R&R, if you have started this chapter in school, do let me know has your teacher covered product of three vectors like scalar triple product and vector triple product? Yes, no. No, sir. No. Okay. So this requires a bit of detailing. So we have to be very careful when we are doing this part. So under triple products or product of three vectors, we'll be talking about scalar triple product, STP or box product, what we call it, okay? And we are going to talk about vector triple product, okay? One thing I forgot, which I was supposed to do after this, so let me re-number it. Next is the concept, second one is a concept of collinearity and coplanarity, collinearity and coplanarity of vectors, including position vectors, okay? This is related to the concept of linearly dependent and independent vectors, okay? Finally, we are going to seal this topic with the idea about vector equations. This is slightly difficult part of it, but it is required for your better understanding of 3D geometry, okay? So introduction is what we are going to start today. Of course, we are also going to touch upon collinearity and coplanarity. And of course, we can start with basic operations also, along with introduction. But yes, product of vectors, that is your dot product, cross product, that will itself take one class, STP, VTP, along with vector equation, that will take another class. So two classes would be definitely required, including the half class that we are left with for today. So introduction, I think vector doesn't need an introduction. We all know that the physical quantities that we have around us are classified either into a scalar quantity and a vector quantity, okay? Now both of them are actually types of tensor quantity. There is something called tensor quantity, which we will learn later on. Tenser quantity is basically when you are trying to use some kind of a multi-dimensional quantity, like a matrix. Matrix is a multi-dimensional quantity, right? So you can say scalar quantity is a kind of a tensor of zero order. Vector quantity is a type of a tensor of first order, because any particular column or any particular row of a matrix will start behaving as a vector, right? So I would not like to get into the complexity of tensor quantity and all, which we will be anyway studying later on, okay? So let us try to understand just the basics of this scalar and vector. So what's the scalar quantity? Since you have all done it in the school, what is the scalar quantity? Something which can be represented by only mentioning its magnitude, right? So they can only be represented by mentioning the magnitude of that quantity. Can you give me some examples of quantities which you know are scalar? Okay, very good. I am getting a lot of responses. Distance, temperature, okay? These are all represented by just mentioning the magnitude, mass, okay? We never mention any kind of a direction while we are mentioning these quantities. On the other hand, what are vector quantities? Quantities that need magnitude, direction, that's it. Now most of you would correct me here and must also follow the triangle or parallelogram law of addition and must also follow the triangle slash parallelogram law of addition, okay? These three things are important. I have seen many people making a mistake that anything which has got magnitude and direction will be called as a vector quantity. Please let me tell you, current is such a quantity which has got magnitude as well as direction but this is not a vector quantity. Why? Because it doesn't follow triangle or parallelogram law of addition. What are these laws which we will discuss it in some time, okay? So this is not going to be a vector quantity because, let me give you an instance here. So let's say I have, let's say a system of wires carrying current like this. Let's say this is 2MP or this is 3MP and I ask you what is the current in this wire? What will you say? You will apply KCL at this node and you will say 2 plus 3 which is 5MP. Now this result change, if I keep the other wire in this way, okay? You will still say that if this is 2MP and this is 3MP, this will be 5MP, yes or no? But if I use the same concept on vectors, let us say I have a force of 2N like this and I have a force of 3N like this, right? And there is one force which I am applying, let's say like this, okay? Which is not allowing this mass, let's say I keep a small particle mass over here. So this force is not allowing this mass to move, right? Or let me write it in this way. I mean you have done your physics to a certain extent, that's why I am asking you this question. So let's say I keep this as 2N itself and I keep a force over here and let's say this angle is 60 degrees, okay? What do you think will be the magnitude of this force such that this particle doesn't move? What will your answer be, okay? You can assume that your x-axis is like this and this is 30-30 each, probably my diagram is not that great here. So let's say I make it exactly symmetrical. So you will say f value will be what? f value will be what? 2N how much, 2 root 3, right? So it will be 2 root 3 in this direction in order to prevent it from moving, correct? Now will this force remain the same if I have the very same arrangement? But now this force is basically acting itself at an angle of let's say 30 degrees from here, right? Or you can reverse the direction if you want. Now if the system is in equilibrium, what is the value of this force? Let's say f dash, right? This value you will say despite everything remaining the same, this value most of you would agree with me will not be the same as the f that you have got over here, okay? This will be of some different nature altogether because here you are dealing with quantities which are vector. They will not follow the scalar algebra, the way the current followed over here. So current does not follow the parallelogram law or triangular law of addition. More particularly polygon law of addition, okay? And this follows the parallelogram law of addition. We'll see what is that and how do we evaluate these answers as of now? This is just from your physical idea of forces that I'm trying to relate it over here. Yeah, yeah, yeah. It seems as if we are sitting in a physics class, okay? So what are the examples of vector quantity? We all know velocity, displacement, acceleration, moment of a force which we call as stock, okay? So these are examples, force, these are all examples of vector quantity, right? Next thing that we would be studying is types of vectors, types of vectors. So many of you are only aware of it, but just a formal introduction of these types of vectors. First is the zero vector. What's a zero vector? Now, even before we start talking about it, we should actually talk about how do we represent a vector? So let's talk about representation of vector, representation of vectors. We represent a vector by either use of a English small alphabet or by the use of a combination of two alphabets with an arrow symbol on top. Many a times in the books, they will not put the arrow, but they will write it in bold, okay? So if A is written in bold or AB is written in bold, that means they are basically trying to represent a vector, okay? So if you have a vector, you can either call it by some name or mention the initial and the terminal points of this vector. So this is called the initial point and this is called the terminal point, initial point and this is called the terminal point. And the way you write the name basically tells you that this is the initial point and this is the terminal point. So if you write BA, BA would have a different meaning altogether where direction will be reversed, okay? So a vector basically has three characteristics. One is magnitude, which we basically represent by the name of the vector written within mod. And the length of this line basically is analogous to the magnitude, okay? And this is shown by the length or indicated by the length of the line segment, by the length of the line segment AB, okay? Now let us say I want to represent 50 kilometers towards north displacement and 25 kilometers towards north displacement. So one vector, if let's say I'm showing it by a 10 centimeter line to show 50 kilometers towards north. And other vector, if I am using to signify 25 kilometers towards north, the norm is we have to draw this vector of half the length. So the length basically tells you in what ratio they are related to each other. Of course, you cannot draw a 50 kilometer long line to show it, okay? Second thing the vector has is the direction that we had already spoken about when we were learning about what are vector quantities. So direction is basically nothing but the orientation of this vector. So it is going from A to B. If I just draw a line segment, line segment doesn't have an orientation. But here we can say that this vector is basically is oriented from A to B. So that is the direction B to A that the the arrow sign will be opposite to what we have shown it over here, okay? The idea of the direction is basically hidden within the unit vector, which is signified by the same vector name. But we don't put an arrow on the top, but we put a cap, okay? So the unit vector basically carries the idea of the direction. So if you want to create a vector, you just do a scalar multiplication of the magnitude with the unit vector. We'll come to unit vector in some time. Third thing that we talk about is the concept of support, which is missing in mathematics because I'll tell you why. First of all, let me tell you what is support. Support is the line from which the vector has been carved out. The line from which the vector is taken. For example, in this case, if I draw a line like this line, as you all know, is an infinitely extending geometrical figure. This vector is basically carved out from this line. Then this line will be called as the support, okay? Now, many a times this word support is not seen anywhere in vectors. For the simple reason is because we deal with something which we call as free vectors. Now, there are two types of vectors. So when I'll be talking about vectors, I'll be talking about free vectors and localized vectors. So free vectors are those vectors which are free to move in space parallel to itself without changing its magnitude and orientation or direction. Okay, those are called free vectors. I'll give an example of free vectors as you know, displacement, let's say. So if I say my displacement is 50 kilometers towards North, okay? Now, will this vector change if I start from my house or if I start from your house towards North? No, right? So let's say this is my house, okay? And let's say this is your house. If I'm going towards 50 kilometers towards North, whether I move from your house or my house and go towards North, this vector will still be called as going towards 50 kilometers North, which is 50J. 50J cap as you all know your IJK confidence. So this is an example of a free vector because it is not localized anywhere in space. Okay, so this vector, let's say if it is a free vector, you can move it parallel to itself anywhere. For example, let's say I draw it here or I draw it like this or I draw it like this. They will all be called a vector only, right? So in maths, we are mostly dealing with free vectors. So in maths, we are going to talk about free vectors and that's why support becomes irrelevant for us because it doesn't matter to which support they fall. So if they are on the parallel support also and having the same length and direction, they will be all the same vectors. On the other hand, there is something called localized vector. Localized vector is basically a vector which is fixed in space. Okay, for example, the weight force of a body. So let's say there's a body of mass m, right? The weight force will always be attached to the body and coming down towards the center of the earth from its center of mass. You cannot draw a weight force of a body like this. Okay, this is your mass m, this is your weight force. This would be wrong, right? Tension, right? For example, let's say there's a mass, there's a block of mass m hanging from a string. So the tension will always be directed away from the body and along the string, right? Of course, on the wall, it will have an opposite impact by Newton's third law. So I can only draw the tension along the string. So this is localized, right? I cannot draw a tension like this, right? Mass is here, tension is here. I cannot draw it. That would be wrong. This is mostly a subject matter of physics, right? So in maths, the vectors are free. In physics, the vectors are not free many a times, okay? Because we mostly talk about weight forces, normal reaction, tension. So they all make the vectors localized, right? Okay, what is the use of support? No use of support in maths at all. There's no support of support in maths. Yes sir, yes sir. Support is just an understanding that, see, support will make a difference when we are talking about, let's say, a stock by a given force at a point. If you change the force, you can say support, of course, R will get changed, right? So probably a force of the same magnitude on a parallel support will generate more or less torque on that, about that particular point, okay? So that, in order to explain those concepts, really support makes more sense. But in maths, it really doesn't make any sense to talk about support. That is why many a times, people have not even heard of the name of support of a vector. What is the support of a vector? Half the people don't know because they only heard, okay, magnitude, direction, magnitude, direction, and probably, yes, parallel grand law of addition. But vector has got a third thing also, which is support. Which is missing because it is hardly used in our understanding of vectors. Anyways, we'll not talk about types of vectors. So I missed out on that, types of vectors. So first we'll start with zero vector. As somebody rightly pointed out, a zero vector is a vector whose magnitude is zero. So if A is a zero vector, by the way, when you're writing as a zero vector, you have to put a zero with a cap on the, sorry, arrow on the top. Sorry, a lot of sound was coming from outside, yeah. So zero vector is a vector which is having magnitude as zero. And direction of this vector is undefined. Direction is not defined, okay? For example, if somebody says my displacement was a zero vector, you cannot comment upon the direction of it. It is just like the argument of a complex number zero, or it is just like degree of a zero polynomial, right? They are all not defined, okay? But at the same time, zero polynomial is a polynomial. Zero complex number is a complex number. In the same way, zero can also be a vector, okay? Unit vector, let me write it in white, unit vector. Unit vector is basically a vector which is along the direction of a given vector. So we normally say unit vector along the direction of a vector. So normally we read it along with the word along a vector. So unit vector along a vector, let's say unit vector along A is basically A cap. As we already discussed this when I was talking about how do we, what are the various parameters of a vector? Now unit vector along A is basically A cap. A cap can be written by A vector divided by its magnitude because we already saw that the unit vector carries the information about the direction, okay? Few famous unit vectors that you will come to know is i, j and k vectors. So basically these are unit vectors along, let me write it like this, unit vectors along positive x axis. So unit vector along positive x axis is basically i cap. Unit vector along positive y axis is j cap. Unit vector along positive z axis is k cap, okay? Next concept is, our next type of vector is negative of a vector. Actually this is not a type of vector. You can say it as a scalar multiplication of minus one with a vector. So let's say if I have a vector A and I do scalar multiplication of that vector with a minus one, I'll end up getting a minus A vector. What is the meaning of minus A? The meaning of minus A is a vector which is having the same magnitude, has same magnitude as A, but direction is opposite. Magnitude as A, but direction is, opposite to that of A, okay? For example, if I say A vector represents 50 kilometers towards north, then minus A will represent 50 kilometers towards south, okay? These are all things you already know. I'm just quickly revising it for you. Next is co-initial vectors. I think this was the third one or fourth one. Which number was it? Fourth one, right? Co-initial vectors. So what are co-initial vectors? Vectors having the same initial point, okay? For example, let's say I have these three vectors ABC. So if they have the same initial point, they will be called as co-initial vectors. Now, if you want, you can make any number of vectors co-initial because we are dealing with three vectors. So even if a vector A was like this, okay? And B was like this. And let's say C was like this. You, if you want, you can make it as co-initial. So if there are three vectors, we can make them co-initial by bringing them in such a way that they are shifted parallel to themselves without changing the direction. So if I just move it a little bit up and back, okay, so it can come to this position, okay? So your B can come to this position. Similarly, if I move this a little bit up and back, then C can come to this position, okay? So normally this co-initial vectors can always be made by you if you have been given any number of vectors. We use this concept in order to use our parallelogram law of addition, which we'll discuss in some time. So if I use the word make them co-initial vectors, then you should know what operation basically I'm hinting at. Next is co-terminus vectors. As the word signifies, basically, they are all terminating at the same point. Co-terminus vectors. So they are all terminating at the same point. Let's say like this, okay? A, B, C vectors all terminating at the same point. By the way, one more thing I would like to add over here. When you want to find the angle between two vectors, you have to make them co-initial. I hope you all know that through your physics knowledge. So in order to find the angle, or you can say angle between two vectors, angle between two vectors is basically defined as the shortest angle, shortest angle between the vectors, between the vectors when they are made co-initial. So that is another important thing that is attached to your co-initial vectors, okay? Next is, next type of vectors that we'll talk about is equal vectors, okay? Equal vectors basically have the same magnitude and same direction, okay? So if these two conditions are satisfied, then these two vectors will be called as equal vectors, okay? Now don't confuse it with something which we call as like vectors. So I'll just write that down also, like vectors. Like vectors are two vectors which have the same direction. Their magnitude need not be the same. So if I say A vector and B vector are like vectors, it means their direction is the same, right? Their magnitudes need not be the same. So you can say equal vectors are types of like vectors. To give you an example of equal vectors, for example, this and this, they're equal vectors. If their length is same, and of course I've made the direction also the same. But something like this will be like vectors, this and let's say this, okay? Then these would be your like vectors, okay? These are your equal vectors. So what is the unlike vector? Let's take it as eighth type of vectors. What is the unlike vector? Unlike vector is where the direction is opposite. So let me give you an example. Let me give you the definition first. So A and B are unlike vectors. A and B are unlike vectors. If the direction of A and B are exactly the opposite, okay? Example, a vector like this and a vector like this, they are unlike vectors, okay? However, please note that they must lie on same or parallel support. So whether you talk about like or unlike, they must lie on same or parallel support, okay? Next is collinear vectors. This is a subject matter of more study. I mean, I'll talk about it in more details because this is a very important concept not only for your, I don't know whether in school it has been taught. Has it been taught in school, NPS, R and R? No? Has a teacher taught you collinear vectors and all? Definitely she would have taught you. Yeah, Anjali is saying yes. Exactly, it is. Okay. Okay, anyways. So basically, like and unlike vectors, they together come under the category of collinear vectors, right? It just means that these two vectors, so let's say if I say A and B vectors are collinear or A is collinear to B. It just means, it just means A and B lie on same or parallel support. In other words, it is a broader category of like and unlike vectors. So let me draw it as a, bend diagram kind of a thing. So let's say this is a set of collinear vectors, okay? Under collinear vectors, you have a set of like vectors and you have a set of unlike vectors. So you can say like vectors and unlike vectors. And within like vectors, you have a set of equal vectors, something like this, okay? It's a very important concept which we'll discuss in more detail, okay? We have to talk about collinearity of two vectors, collinearity of three points. Now, many people ask me, sir, why do we call it collinear? Because they may lie on parallel support also. Collinear means having the same line, okay? Right? Collinear. So they need not lie on the same support. So why do we call it as a collinear? It is because our vectors are free vectors. Let us not forget it, okay? So even if let's say your A vector is like this and your B vector is lying on a parallel support like this, you can always translate B parallel to itself and bring it on the same support as that of A. That means you can bring it in line with A. In other words, you are making it collinear to A, okay? So thanks to the concept of free vectors that even for parallel support of two vectors, yes, their magnitude, their direction, their magnitude need not be the same. Direction may be anti-parallel also, parallel or anti-parallel also. Both the conditions are basically considered under collinearity of vectors only. We'll talk about it in more detail. So we'll talk about in detail later. Details would be later, taken up later. Next is your coplanar vectors. So what are coplanar vectors? Two vectors, first of all, are always coplanar. So we always talk about coplanarity of three vectors. So we say three vectors, three vectors, A, B, C are coplanar, are coplanar. When they lie on same plane or parallel planes, okay? Now why parallel planes? Because we are dealing with three vectors. So I can always bring one to the plane of the other. Okay, now I'm trying to understand this. Two vectors, okay? Whether they are skew or not skew. Now what is the meaning of skew and not skew? How do I explain you? Let's say this is a vector which is coming out of the, you know, coming out of the plane like this. That is coming towards my face. And this yellow line is on the plane of my screen, right? So you can imagine as if there's a flyover and there is a road passing below the flyover, right? So we say these two are skew actually. They are called skew, okay? So even if two vectors are skew or non skew, they can always be made coplanar because of the simple fact that we are using three vectors in max. So this flyover can be brought down to the level of the road. Okay, so you can bring it down on the same plane. Are you getting my point? So imagine like, you know, how do I explain it to you? Imagine, let's say this is a plane which has one vector. Let's say this one, okay? And there's another vector like this. Let me put it in white, okay? So you can always translate this guy. Let's say this is your A and this is your B. You can always translate this B and bring it to the plane of A, isn't it? So you can, two vectors are always coplanar. Let me write it down here. Two vectors are always coplanar because we are dealing with three vectors, okay? That is why I started with three vectors, not two because it is needless to talk about three vectors or two vectors coplanarity because they're always coplanar. Now three vectors may not be in the same plane. For example, your I, J and K. No matter whatever you do, whatever translation you do on these vectors, you cannot bring them on the same plane. One of them will always be perpendicular to the plane which contains the other two. So they will be called as a non-coplanar vectors. But if there is a case where you can bring the three vectors in the same plane, right? By of course, if required, translating them parallel to them's initial direction, then they basically will be called as coplanar vectors. This again, we'll talk in detail later on, okay? Details will be shared later on on this. So we are good to now take up the next one which is basically the concept of addition of vectors. There is one more type of vector which we'll talk about after this which is called position vector. So first we'll learn addition of vectors. Now I'll be fast here because you are already aware of it. I don't think so. There's anybody who doesn't know how to add two vectors. So addition of vectors basically, what we call as the sum of vectors or resultant of vectors, resultant of vectors. So if there are two vectors given to you, let's say A and let's say B, okay? And you want to add these two vectors. First of all, let us understand what is the meaning of it. When you're adding these two vectors, basically you are trying to replace them with such a vector. Let me draw it in a different color. Let's say blue color. You're trying to replace it with such a vector. Let's say, yeah, I'm just drawing it like this, okay? Like you're trying to replace it with such a vector which does the same role as A and B together will produce in that system, on that system. For example, you're talking about resultant of two forces. So resultant of two forces such a force which will create the same amount of, you can say, effect with those two forces in combination will produce, isn't it? So when you are finding the resultant of two vectors, we normally use two types of laws. In fact, we can say three types of laws. One is called the triangular law of addition. So let me show you how this triangular law of triangle law of addition works. When triangle law of addition is applied to find the sum of two vectors, we normally connect the terminal point of one with the initial point of the other. So what do we do is, let me just draw the same structure. A, I connected to the terminal point of A, I connected to the initial point of B. So let's say B was like this. Note that I have drawn it in such a way that I have kept A here and A here in the same direction and of the same length. B also have drawn it in the same way. So now this resultant vector would be a vector which will connect the initial point of A to the terminal point of B. This is basically what we use in order to find out the resultant of two vectors. Now, a broader version of triangle law of addition is the polygon law of addition. By the way, when you are trying to show the resultant of two vectors, we normally make the resultant with a dotted line. We normally make it with a dotted line showing that this is the resultant of these two vectors. That means all the three are not present at the same time. Either A and B combination is there or this dotted line is there. So don't think like, these are three vectors operating like this, no. A and B resultant is this. So either you will show C or you will show A and B together because both will result into the same effect on that physical system. Now what is polygon law of addition? Polygon law of addition is just an advance or you can say extended version of triangle law of addition. So if you have been given three vectors, let's say, or four vectors, I mean n number of vectors, it'll work for any number of vectors. So let's say these are your three vectors, A, B and C. Okay. And now you want to add them. So we'll follow the same approach as we did for the triangle law of addition. We'll connect the terminal point of one with the initial point of the other. Okay. So this will be connected to this. This will be connected to this. Okay. So when you do that, let us make a fresh diagram. A, B, and let's say C. Note that I have made them, I've made these arrows in such a way that I have kept them parallel to this. So I'm not changing the direction and the length. Okay. Now a vector which connects the initial point of A to the terminal point of C, this would be your A plus B plus C vector. Okay. Few things to be kept in mind over here. It doesn't matter in which order you are connecting it. For example, let us say I connect the terminal point of B with the initial point of A and then initial terminal point of A with the initial point of C. The resultant will still be the same length and a direction parallel to this. It is not going to change. So order of vector addition doesn't matter. Even if I do this, it doesn't matter. Even if I do this, it doesn't matter. Okay. Okay. So you can say commutative law basically holds good for vectors. Next is your parallelogram law of addition. Parallelogram law of addition makes more sense when your vectors are localized. Especially when you are, let's say, talking about two forces acting on the same body. Or let's say a body is connected by multiple strings. Okay. There you cannot, you know, shift these forces. Of course, I mean, you can always apply. But in order to know your exact direction, because you don't want to shift them because it will change your view or orientation at which you're looking at the diagram. So their parallelogram law of addition makes more sense. So parallelogram law of addition says when you want to add two vectors, make them, make them co-initial. Okay. So what do we do? We are now going to make these two vectors. Let's say we want to add them. Okay. So we will make them co-initial. That means something like this. Please note I'm not fiddling with the length and the direction of A and B. They have been just moved parallel to themselves. So parallelogram law of addition says once you have made them co-initial, you complete a parallelogram with A and B as the adjacent sides. Okay. So once you have done, then make a vector connecting this co-initial point to the opposite vertex over here. This will become the resultant of A and B. Of course, you need not even show this, but I've shown it to you. So this will become your A plus B. Okay. And this result will not change because if you see it is basically the same thing that we did for triangle law. If you shift this B to this position, it is basically performing the same functionality as what the triangle law was performed. Is this fine? I hope all of you are aware of this. In a similar way, if I talk about subtraction of two vectors, subtraction is basically nothing but adding a vector to the negative of the other vectors. So concept remains the same. So but I'll show you one instance of that. Where should I show you? Where should I show you? Let me show you over here. So let us say A vector is this and B vector is let's say like this. Okay. And I want to do A minus B. What should I get? Okay. A minus B is to be treated as A plus negative B. Negative B means you will just reverse the direction of B. So let's say A is like this. Okay. B direction will now be reversed like this. Okay. Now, which law you want to apply? You can apply it. Triangle law, you can apply it. You can connect the terminal of one with the initial of the other. Okay. So let's say I connect this point to this point. So what I'll get something like this. So let's say I'm adding these two. So this will be connected to the initial of this. Okay. And then make a line connecting the initial point of B with the terminal point of A. Okay. So this will be your A vector. This will be, let me name it. So this will be your A minus B vector. Okay. You may use your parallelogram also by making them co-initial. One thing that I would like to highlight, if you're using parallelogram law. So let's say this was your A and this was your B. Okay. When you connected or when you completed a parallelogram, we discussed that this vector represents A plus B. This vector represents A plus B. Correct. Now, let me show A minus B on the same diagram. So A minus B, what you will do, you will just reverse the direction of B where you will say like this. Sorry. Yeah. This is minus B. And since you have to find the resultant of A and minus B, you will complete a parallelogram like this. Correct. And then you will make an arrow starting from this point all the way to the opposite vertex and you'll say this is A minus B. Correct. Now the same length line is actually and the same direction line is actually the line that you would get when you connect the other diagonal. So if you connect the other diagonal of this parallelogram, then this particular vector will actually be A minus B. So you don't need another parallelogram to know what is your A minus B. You can do the same thing on the parallelogram which you use for addition. Okay. But be careful about the orientation. Many people draw it like this. Now how do I know in which direction to apply the other, which direction to make on the other diagonal? See, if you want A minus B, then move in the direction opposite to B and then in the direction of A. So basically you're moving from here to here. So this will be the direction of the arrow. If you're making it in this way, which is basically wrong, then basically you're trying to find minus A plus B. So this will be minus A plus B. Okay, be careful about it. I have a question for all of you before we go for a break. Let us say there is a regular hexagon. This is the regular hexagon. Regular hexagon means the side lengths are all same. Okay. Let's say this is vector A. This is vector B. Let me name them. Name the vertices A, B, C, D, E, F. Okay. My question is, find in terms of A and B, the following vectors, C, D, E, F, F, A. Only in terms of A and B, we have to write these vectors. Yes. Who will tell me C, D vector? Because I think D is very easy. What is D? D is minus A. I'm sure you would have got it. Okay. So C, D vector, I'm getting answers from few of you. Aditya, Ravi Kiran, Sharmik, very good. Anybody else? E, F, E, F is easy. E, F is negative of B, C. So I can call it as minus B. So E, F is a negative of B, C vector. Okay. Oh, sorry, not here. Here, down. Yeah. What is C, D, and what is F, A? Of course, you know, C, D, and F, A will be opposite of each other because they are exactly anti-parallel of the same length. So if I get one, I would get the other one also. So how do I get that? Okay, very good. Okay. Now all of you please pay attention here. If I connect A to D and call it as a vector, let me call, connect A to D. What will be AD vector? Now AD vector would be a vector which is parallel to BC vector, okay? And not only that, its length will be double that of BC. So AD length will be two times of BC, isn't it? So can I say AD vector is 2B vector? Now don't ask me how it is double. It is just a simple trigonometry can tell you how it is double. This is basically if you drop a perpendicular from here to here. Then this length is also the same as length. Let's say this length is X and this will also be X. And this is 60 degree. This is also of X length, so this will be X by two. This is 60 degree, so this will also be X by two. So X by two, X, X by two will be two X. So double the length. So this will be 2B vector. Now if I use my polygon law of addition, can I say A plus B plus X, not X, let me call it as CD vector, which we don't know. Can I say if I go from A to B, B to C, C to D, it is as good as going from A to D. That means AD is the resultant of AB plus BC plus CD. So CD is my subject of the formula over here. So can I say CD would be B minus A. So CD vector would be B minus A. FA vector would be A minus B. Okay, very easy. So basically I use my polygon law of addition to get this job done. Absolutely, absolutely Kirtanam. Okay, so I'll start with position vector but only after a small break. Welcome back after the break. I hope everybody's back. Okay, except Venkat, everybody's back. Okay, anyways. So having known the concept of addition of vectors, we are now going to talk about something very, very important which is position vector. Okay, position vector is that concept in vectors which links it to coordinate geometry. So whatever fundamentals you have learned on coordinate geometry, you can apply them to vectors provided you are using this concept of position vectors. So first of all, what is position vector? So position vector of a point. Position vector is always for a point. So let's say if I say position vector, position vector of a point A is let's say A. What does it mean? It means if this point A in space, okay, I connect it to a reference vector like this. So I make a reference vector, a reference point. I can say O and then make a vector connecting O to A. Then this vector is what we call as the position vector of point A. So many a times we would not show this origin or reference point and this line. We directly will say there's a point A whose position vector is A. Automatically you can choose your reference point to be anywhere in space and imagine that there is a vector connecting that reference point to A starting from reference point and going towards A. It is very similar to stating the coordinates of a point. When you say there's a point whose coordinate is two comma three. We never mentioned the origin, right? It is up to you to choose an origin. It is up to you to choose the reference lines and hence the reference point and then choose the point two comma three, right? So when coordinate geometry questions are given to you, they say there is a point with coordinate one comma five. Do they mention where is the origin? No, right? It is up to you to choose the origin and accordingly make your one comma five point. In the same way, when we talk about position vector of a point, they will just say there's a point whose position vector is so and so. Let's say there's a point A whose position vector is A. They would not show this. This is up to you to take, okay? Will the results change if you take somewhere, the position vector somewhere else? No. It is just like saying, if I take a point two comma three by choosing the origin over here or if I take a point two comma three, like choosing an origin over here, are there two different points? Yes, of course, there are two different points in space, but we only deal with the relative positions, not with the absolute positions, okay? So position vector can be treated as if you are talking about coordinates of a point. Now, what are the benefit of position vector? What is the application of position vector for us? Position vector helps you to apply section formula. You can call it as utility of, or use of position vector concept, use of position vector concept. It basically helps you to apply your section formula that you have already learned in your coordinate geometry. So before we start, let me just take a point A, okay? Let me take another point B, okay? This point, let's say has position vector A. This point, let's say has position vector B, okay? Now, let us say I want to find out the position vector of a point C. There's a position, there's a point C on this line, okay? Whose position vector is C? And I want to find out the position vector of this point C which divides this in the ratio of M is to N. Let's say internally, as of now. How will you find it out? So what is C in terms of A, B, M, and N? How will you find this? Now, let us try to understand this concept. First of all, I would like to know from you if you know the position vector of A and the position vector of B, what is AB vector? What is AB vector? B minus A, how do you know that? Starts from A and goes to B. So that makes it B minus A, huh? Starts from A goes to B, so by what logic does it become B minus A? Of course, answer is correct, but why B minus A? Okay, let us try to understand this from our addition of vectors concept. Let's say I take the reference point to be here. I can take anywhere I want, okay? When I say position vector of A is A, it means that if I connect O and A, then this vector is your A vector, correct? By very basic definition of position vector. And if I connect B to point O, then this will become your B vector, correct? Now, if I want to find out what is AB vector, I can use my addition of vectors concept over here. So let's say if I start my journey from O to A and then A to B, can I say it is as good as saying, I have gone from O to B, yes or no? Right? O to A is A vector, AB is unknown to me and O to B is B vector. So from here I can make AB as B minus A. Now, this is a very important thing that you all need to note down. Please note, when you know the position vector of two points and you want to know a vector connecting those two points, so it is always the position vector of the destination minus position vector of the force. So this is something that you need to commit to your memory because this will help you save a lot of time. So position vector of A and B if they are known, then AB vector will be position vector of B minus position vector of A. Okay, now having known that, my problem statement is to know this guy. My problem statement is to know this guy because this guy is what is your C vector, isn't it? Okay, let me draw B outside, else we'll get confused. Yeah, so how will I come to know that? Now try to understand this fact that AC vector, AC vector will be collinear to AB vector, okay? So AC vector is collinear to AB vector, okay? You can say AC and AB are like vectors, also you can say that, okay? Now when two vectors are collinear, I think we have discussed this very briefly that one can be expressed as scalar multiple times the other. Correct? In this case, we can easily know the scalar quantity because we know that the length of AC is m by m plus n times the length of AB. So I can say lambda is m by m plus n times the length of AB. Is this fine? Any questions here? Okay, now if I know AC vector, can I use my again addition law vector addition that OA plus AC is equal to OC? So can I say OA plus AC is equal to OC? Correct? So OA is A, AC is m, AB vector, sorry, m by m plus n times AB vector. AB vector just now you told me it's B minus A, okay? So if you simplify this, you would realize that you have actually got something very similar to what you had seen in your section formula. This is your OC vector, OC vector is actually a C vector. So this one gets canceled, so it becomes MB plus NA by m plus m. Yes or no? This becomes your position vector of C. So closely resembles to that of a section formula. So a lot of coordinate geometry problems and a lot of vector problems can be solved by a mix and match of these concepts. So now I have got a small question for you all. By the way, everything is clear over here, any explanation anybody wants? So is it clear what is the position vector of a point? Again I'll repeat it, position vector of a point is a vector connecting the reference point to that particular point directed from reference to that point. It is analogous to your coordinate of that point in coordinate geometry. The use of it is we can use our coordinate geometry fundamentals to solve our vector questions as well. Okay, so here I have one question for all of you. O is the circum center which they have actually assumed to be the reference vector also. O dash is the ortho center of a triangle ABC. Prove that O A, O B, O C, then of course, O A, O B, O C will represent your position vectors of the vertices respectively is equal to O dash. Anurag, your mic is on. I'll meet you, okay? Okay, so let's say this is your ortho center O. Okay, we all know how ortho center is formed. Let's say this is your altitude and somewhere on this altitude there is an O dash, ortho center. First part, anybody is done. If you're done, please put it on the chat box that you are done with the first part. Okay, let me take this up. I think there's no response coming from anybody. Let me drop a perpendicular from O on to BC. So we all know D is the midpoint of B and C, right? Because that's how a circum center is basically obtained by perpendicularly bisecting the sides, right? So if I drop a perpendicular from O on to BC, basically it will meet on D, D being the midpoint of B and C. Okay, now, if I say O D vector is equal to, or you can say two O D vector is equal to A O dash. That means this vector is half of O two B. Sorry, A to O dash, how many of you would agree to this? I'm sure many of you would be getting a doubt, how? Okay, now, first of all, can I say the direction of A O dash and the direction of O D would be the same because both are perpendicular to the same line? That means they are parallel, correct? In other words, they are collinear, right? If I can somehow show that the length A O, if I can show that the length A O is twice of O D, then probably you will be convinced to believe this, right? For that, I would go back to bit of properties of triangles. Okay, so let me draw the same figure again over here. Let me draw the same figure over here, again over here. A miniature, I mean, scaled down version, okay? Let's say I drop a perpendicular. And this is your ortho-center, okay? Now, first thing that I would like you to tell me, can I say this is your radius of the circum-center R? Can I say this angle, this angle would exactly be this angle? Are you convinced with this first of all? So this entire angle was A, this entire angle was A. So this angle will also be A. Do I need to explain how? Yes, sir, please, could you please explain? Okay, so then I'll take you even further back to your class 10th days that the angle subtended here, if it is A, then this will be two A. And if I drop a perpendicular, then both will be A, A each. Now clear? Oh, yes, sir. Thank God, okay. So if this is R and this angle is A, can I say OD length would be two, sorry, OD length will be R cos A, basic trigonometry, correct? Now, let's say this was your ortho-center. Let's say this was your ortho-center, okay? Let me call it as H as of now, or let me call it O dash only, okay? And let me connect like this. So basically you know this is 90 degrees, okay? Now, in a triangle, you all know the convention of naming. This is side C, right? And this is angle A, so how much is this way? If this angle is A, this side is C, how much is this length? Let's say O, M, I call it. How much is A, sorry, how much is A, N? Obviously you'll say C cos A, correct? Now, if this angle is C and this angle is 90 degree, can I say this angle will be 90 minus C? In 90 minus C or a triangle, you see this short two triangle over here, okay? This base is C cos A. This angle is 90 minus C, so how much is the hypotenuse? So you'll obviously say C cos A, let's say hypotenuse is A O dash, okay? So C cos A by O dash will be equal to cos of 90 minus C, which is sin C, correct? In other words, C by sin C cos A is equal to A O dash. C by sin C is known to be two R from our sin rule. What did you see here? That your A O dash is two R cos A. So what I'm trying to claim here is that, this length is double of this length and since they are parallel to each other, I can easily say that A O dash vector will be double of O dash. Now is this how answered? Yes, sir. Okay. So how is answered, but how does it help us to solve the question? Let us look into the question here. Question is that I have to prove that A O, sorry, O A, O B, O C is O O dash. O O dash, let me connect. Okay, so what happened to my control of the pen? This is O dash. Okay, now let us try to understand this. So from the figure, can I say, from the figure, can I say A plus A O dash is equal to O O dash. Basically I've used vector addition. So if you go from O to A and then come from A to O dash, basically you have traveled from O to O dash. Right, and just now we figured out that A O dash was to O D vector. Now, come back here. Come back to the fact that D is a midpoint of B and C. If this vector is basically position vector B and this vector is position vector C, can I say D's position vector will be B plus C by two section formula, just now we learned. Okay. So can I say, O D vector is B plus C by two. That means two O D vector is B plus C. And B is nothing but O B vector and C is nothing but O C vector. In other words, this two O D I'm going to replace with O B plus O C and that is what we wanted to prove. Got it. Is the story clear after this step? How is two O D becoming O B plus O C? So O D is nothing but, see, reference to that point is nothing but the position vector of that point, right? So O D vector is B plus C by two, where B is the position vector of B point and C is the position vector of C point. So O D is B plus C by two means two O D is B plus C and B itself is O B vector, C itself is O C vector. So I'm replacing my two O D with O B plus O C. Makes sense. So this was a deep problem actually because you needed to know your, the fact that this is double of this length and that will only happen when you know your properties of triangles very well. Most of you I guess would have long forgotten all those concepts. Okay. Next part of the question. O dash A plus O dash B plus O dash C is two O dash O. Let's do one thing. Let's go to the next page because this is totally cluttered up. Even if I want to write something, I will not be able to write. So I'll give you the question in one second. Just type on the chat box if you're done with the second part. This is called E, let's use all the alphabets A, B, C, D. Yes. Anybody done with this second part? See guys, you're not applying your mind. What can you say about O B, O dash B and O dash C? Can I say this is actually two O dash D? No, why? So for the simple reason, now you take this as your reference point. Correct? Yes or no? So now this position vector, they are not going to change with respect to this point. Let's say B's position vector that is O dash B is your B, O dash C is your C, then D will be B plus C by two. Correct? Right? So basically now I'm asking you to change your reference point to O dash. And as I told you, it is up to you where you want to choose your reference point. I can shift my reference point as I want to. Okay? And I can call. Okay, let me not call this as O A now. Now this will be called as O dash B now. So your B vector would be nothing but O dash B. Your C vector will now be known as O dash C. And your D vector, which is actually B plus C by two will now be known as O dash D. So what I'm trying to say here is that two O dash D is B plus C. That means two O dash D is equal to O dash B plus O dash C. Correct? So the term that I have over here in the given expression, the last two terms can be written as two O dash D. Okay, so now let me apply it. So in this particular expression, I'm going to replace these two guys with two O dash D. So I'll end up getting O dash A plus two O dash D. Now O dash A is something that we just now figured out was two, was two? O D. O D. Okay, but that will not solve our purpose. Okay, so let me do one more trick. Let's, let's, let's, let's. No, okay, yeah, that's all our purpose. No, why not? Yeah, it solves our purpose. So I can, sorry. So I can write this as two, once again, O dash A is Ulcha, right? So it is negative O D, right? Yes or no? Venkat. Yes. O dash was two O D. So I'm not talking about O dash A. So it is negative two O D. Oh, yes. Okay. So now we can write this as plus O dash D, which is actually two O dash D plus DO. And O dash D plus DO is nothing but O dash O. Is this what they wanted us to prove? Two times O dash, yeah, two times O dash. That's good. Okay, so you can see for yourself, if you move from D to O, if you move from D to O and move from O dash to D, O dash to D, and then you move from D to O, basically you have gone from O dash to O. Okay, so this is how you go. So you're moving from O dash to D and then you're moving from D to O. So basically you have come from this way to this way. So O dash to O. So it is two times O dash O. Clear? Any questions, any concerns? Any questions? Sir? Yes, sir. Sir, we can write O, O A plus A O dash is equal to O dash, no sir. Now, once again, repeat again. O A plus A O dash is equal to O dash. Yes. Sir, and then from that we get O dash A is equal to O A plus O dash O. Like if we do some work, you know. O A is equal to O dash O dash A. Sir, what I was trying to do is basically, since we know that O A plus O B plus O C is equal to O dash, and then we can get O dash A in terms of O A and O dash O. And believe them, right? O dash B as O dash O plus O B as something. Yeah, and then we just add them then it will become three O dash O plus O dash. Okay. Okay. This is a little shorter, sir. Yeah, yeah. Even that is fine. Good, good, good. Nice. Let us now solve the third part. Solve the third part. A O dash O dash B O dash C is two A O. Forget about A P right now, just prove two A O. They're just trying to say P is a point which is on the diameter. A P is the diameter through A of the circle. So if you basically complete a circle, basically I'm trying a miniature version of it. So had your figure been like this, then A P is basically a diameter like this. So this is your P point, what they're trying to say. Of course O will be there. Yeah. So how do you do that? A O dash, A O dash, O dash B O dash C is two A O. You can use the previous result I think. And you've already done all the hard work. One second. Sure. Just type done if you're done. Go slightly down. Very good, Tipan. So see, for the last part, you are basically looking for A O dash plus O dash B plus O dash C, right? What I'll do here is, I'll write A O dash as two A O dash minus A O dash. Okay. What about the vector notations on top? Rest of the terms, I'm not changing it. Let me write them as it is. Now A O dash, if I remove the negative sign, we all know that it will become plus O dash A. And this is something which we already derived in result number two, right? What was it? Two O dash O, right? So you have two O dash O. So if you take two common, you'll end up getting A O dash plus O dash O. That means you're going from A to O dash and then O dash to O, which is clearly like saying you are going from, you're going from A to O. So it is two times A to O, which is what we wanted to prove. Okay. And they have basically claimed that, which is actually true also, that this is actually an AP vector. Two times A O is an AP vector, which is two, because AP and A O are collinear and AP length is double the length of A O. So this is actually your AP vector, as per the diagram I have shown you. Is that clear? Any questions here? Anybody? So let's take another one. This one we already did. Okay, let's take this one. Two forces AB and AD are acting on the vertices of a quadrilateral ABCD. And two forces CB and CD at C. Prove that the resultant is given by, that is AB plus AD plus CB plus CD. That is the resultant is given by four EF. E and F are the midpoints of AC and BD respectively. Yeah, everything is free vector here. Anjali, no worries. Let us draw the figure for it. I'm sure everybody has drawn the figure for it. So let us say this is my quadrilateral. AB, BC, CD. Okay, let's call this as A point, B point, C point, and D point, right? So there is a force along AB. There's a force along AD. Okay, there's a force along CB. CB is this direction. CD is this direction. Okay. And they have also talked about or referred to the midpoint of AC and BD. So let's say the midpoint of AC is this. If we call it as E point. And midpoint of BC is this, if we call F point. So basically what they're trying to say is that AB plus AD plus CB plus CD is four times EF. This is what they ask us to prove. Okay, now this problem can be solved very, very easily if you deal in position vectors. So what I'm going to do is I'm going to assign position vectors to these points. Okay. So in terms of position vector, if I start writing AB, AD, CB, CD, let's talk about LHS. What is the AB vector? What had we discussed? When you know the position vector of two points, use always the destination minus source. So yes, absolutely correct. This will be, this will be, this will be okay. So if you collect your terms, you will see you get minus two A, minus two A. You'll get two B, minus two C, and two B. Okay, very good, LHS is this. Now let's talk about RHS. For RHS, I need to know what is position vector of E first of all and what is the position vector of F. Now we know that E is the midpoint of A and C. So I can write E's position vector as A plus C by two by the use of midpoint formula, which is a special case of section formula, and OF, OF means position vector of F is nothing but B plus D by two. Okay. So EF vector is nothing but OF minus OE, isn't it? Yeah. So OF is B plus D by two, OE is A plus C by two. Okay, this is EF vector. Let us multiply both sides with four. So if you simplify this, you yourself will see that you have written minus two A plus two B plus two C, sorry, minus two C, minus two C plus two B. Which is clearly your LHS, sorry, which is clearly your, this expression. Okay. So same, both the sides and hence we can say AB plus AD plus whatever was it, CB plus CD. CB plus CD is four times EF. Okay. So position vector basically gives you a shortcut where you don't have to apply your addition for addition laws or addition of vectors concept. You can directly deal in terms of coordinate section formula and that will make your life super easy. Is that fine? Any question, anybody? Let's take another one. ABCD is a parallelogram. Let me draw it. ABCD is a parallelogram. E is the midpoint of BC and F is the midpoint of CD, okay? AE and AF are connected, okay? And then BD is connected, okay? Using vectors, find DP is to PQ is to QB. Now, if I were you from the look and feel, it looks like one is two, one is two, one. So that's what happens when you draw, I put it figure. But let us do it by use of vectors. Let me use by using vectors. Yes, so P and Q are like points which trisect the join of B and D. A, E and F are midpoints. Yes, Aditya, this is the midpoint. So this is same as this and this is same. Yes, any idea how to do it? One second, one second. No need to, see basically section formula only you can help you. There's no hardcore geometry involved here. It is just vectors, nothing else. Okay, so let us discuss this. See, what I'm going to do is, I'm going to assign position vectors by keeping A as my reference vector, okay? Okay, so let's call this as B. Let's call this as D. And by the fact that it's a parallelogram, can I say C vector, position vector of C will be B plus D. Basically I've used my parallelogram law of addition. So unnecessarily do not use A, B, C, D and all that will just complicate your things. Here I have tried to use minimum amount of variables as I can. So what I've done, I have taken my A as the reference vector. I've called my B position vector as B. I've called my D position vector as D. And automatically by the virtue of the fact that it is making a parallelogram, your C position vector will become C of B plus D. Okay, now let us call or let us find F position vector. F position vector being a midpoint, can I say it is B plus 2D by 2? Similarly, E's position vector will be 2B plus D by 2, correct? Okay, now what I'm going to do is, I'm going to first take this ratio as Dp is to Pb as lambda is to one. So I'm going to call this as lambda and this as one. Okay, similarly I'm going to call this as let's say, we can call it as beta is to one. Okay, this is beta is to one. Now, P as you can see is a point which divides the join of B and D in the ratio lambda is to one. So position vector of P, or you can call it as AP in this case because itself is your reference vector. So can I say position vector of P, that is your AP will be lambda B plus one into D by lambda plus one, right? Correct? And not only that, AP vector is also beta by beta plus one times AF vector and AF vector is B plus 2D by two or you can use section formula if you want. Okay? Now since both of them represent the same position vectors, can I compare them? Okay? So when you compare them, the coefficient of B on both the sides, if you compare you get lambda by lambda plus one is one by two beta by beta plus one. And if you compare the coefficient of D, then one by lambda plus one is beta by beta plus one. Okay? Let us divide one by the other. So what do you end up getting? What do you end up getting? Lambda equals half. Lambda is equal to half, yes? That means this length is exactly half of this, okay? So I can say that this ratio here will be one is to two, right? Yes or no? If you do the same activity with P point, I'm not doing it, but you can do it in a similar way. You would be able to see that this is one is to two. What does it mean? It means that P and Q will be trisecting the join of B and D and in that case this ratio will become one is to one is to one. That means DP length, PQ length and QB length will all be the same. Are you getting my point? So I'm not doing it, but you please prove that similarly. BQ is to QD will also be one is to two, okay? So please do this for your assignment. Is that fine? Any questions? All right. We'll take more such questions, but after we have studied a small concept, which is resolution of vectors, resolution of vectors. So resolving vector is a very, very fundamental way of solving your questions in physics. You must be resolving vectors into perpendicular directions, correct? So let us say you have a 2D vector, two dimensional vector. You can always resolve it along the X and the Y axis. Okay, so let's say I have a vector, which is given by this. Let's say a 2D vector A, okay? Now I have purposely chosen my origin to be at the initial point of A. Okay, now let us say the coordinate of this point becomes A1, A2. So what I'm doing is I'm putting my coordinate axes in such a way that the initial point of a 2D vector, which lies on the plane of my board. Yeah, which lies on the plane of my board basically becomes in such a way that this point is A1, A2. Now, if this is A1, that means we all know that if you project this down, this length will be A1, correct? And if I project it on the Y axis, this length will be A2. So if this is A1 and this is A2, that means this is also A2, correct? Now if I use a vector here, let's say I call this as OM. So if I use a vector here and here, I can clearly see that OA, let me call this point as A point. OA would be the resultant of OM and MA. OM is what? OM is basically a length A1 along the positive X direction. So I can write it as A1, ICAP, because we know that any vector can be generated by multiplying its magnitude with a unit vector in that direction. So unit vector here will be ICAP. Similarly, MA would be nothing but A2, JCAP, right? In other words, what you have done, you have basically resolved the vector A along X and Y axis direction, okay? This is called the concept of resolution of vectors. And one more thing that you'll understand from here is that if you know the position vector of a point, or if you know the coordinates of a point, its position vector, which is actually OA, is nothing but attach I with its X coordinate and attach J with the Y coordinate. So this will become the position vector of A. Got this? Now, if I extend this to three dimension, how do we resolve vectors in 3D? So for three dimension vectors, we need, of course, our right-handed coordinate system, okay? And let us say I have a vector OP like this. And let's say this point is A1, A2, A3. Now, in order to reach this point, if you just take a projection of OP on the X axis, guys, get this correct over here. You are basically making the X axis and OP come in the same plane, okay? Once you make them come in the same plane, then you're dropping a perpendicular. So this 90 degrees that you see is in the plane which contains OP and OX. Are you getting my point? See, it is in 3D dimension, right? So imagine that there is a plane which passes through OP and OX, okay? So that will be slightly a tilted plane, right? Inclined plane. On that plane, I drop a perpendicular from O on to this. This angle is 90 degree, right? And this length will become A1. Similarly, if I drop a perpendicular on to OY, okay? This length will be basically your A2 length. This is what we have learned in our basic 3D, or you can say introduction to 3D in class 11. And again, if I drop a perpendicular here, this length will be your A3 length, okay? So in order to move, in order to go from O to P, I can take a path like this. I'll move like this. Let's say O to M. Then I will move in this way, okay? Parallel to this. So basically I'm moving along, let's say MN direction, okay? And then from this point, I've actually gone further. Yeah, okay? And from this point, I decide to go up like this. Okay? So probably my perpendiculars are not that, this thing it will be going in this direction actually. Anyways, so what I basically want to say that, you are moving I units in the direction, you are moving A1 units in the direction of I, A2 units along the direction of J, and A3 units in the direction of K. So your A would be the resultant of these three small vectors. So this vector plus this vector plus this vector, okay? So according to your polygon law of addition, this plus this plus you go up, that will be same as going from O to P. In other words, if you know the position vector of a point, then a vector connecting origin to that point, sorry, if you know the coordinates of that point, then the position vector connecting origin to that point is just obtained by putting I, J and K unit vectors next to their X coordinate, Y coordinate and the Z coordinate. Okay? When we go further, we will understand something very interesting over here. With respect to 2D vectors, if somebody says what is the, if you know the resolution of the vectors and somebody says what is the direction of that vector? Okay? Then I'm sure you would say it is tan inverse A2 by A1. Okay? So this angle theta is tan inverse A2 by A1 with the X axis, correct? How do you mention the direction of a 3D vector? Now later on we'll study this in more detail, but as of now I'm giving you a very, very brief idea about how do we talk about direction of a 3D vector? Yes, yes, yes, correct, correct vector. So if somebody says there's a 3D vector OP, okay? What is the direction of OP vector? How will you mention it? Let's say they have given that OP is, OP is A1I, A2J, A3K, okay? So for the 3D vector, how do you find out the direction? Okay? Now in 3D vectors there is actually direction cosines, okay? Because now you cannot mention its angle with respect to any two coordinate axes. You have to mention its direction with respect to three coordinate axes. So what do we do? We basically call this angle as alpha, this angle as beta, and this angle as gamma. Note that these angles are basically on the plane containing the vector and the respective axes. For example, let's say OP and X axis. When you bring it on the same plane, let's say they are on the same plane, okay? Then the angle on that plane is alpha. Similarly, if you bring OI and OP on the same plane, then the angle here will be beta like that, okay? So now in case of a 3D vector, you basically talk about cos alpha, cos beta, and cos gamma. You need three things to tell its direction, okay? And cos alpha is basically nothing but A1 by A1 square, A2 square, A3 square under root. Cos beta is nothing but A2 by under root of A1 square, A2 square, A3 square under root. And cos gamma is nothing but A3 by under root. Under root of A1 square, A2 square, A3 square, okay? So these are the direction cosines, cosines of a 3D vector. We will talk about this in more detail. By the way, these are called LMN. I think with Nafla's I started this today itself, correct? Skanda? Shankin? Yes, sir. Okay, so we'll talk more about it when I do 3D geometry with you, okay? So this is resolving of vectors. Now let us take few questions on this. Okay, let's start with this question. This is a pure physics question, a relative velocity question actually. A man traveling towards the east at eight kilometers per hour finds that the wind seems to blow directly from the north. On doubling the speed, he finds that it appears to come from northeast. Find the velocity of the wind, done? Anybody? Assume things are in 2D only, okay? Very good, Venkat, very nice. Others, okay. So basically what has been provided to us is that if the man is traveling towards east at eight kilometers per hour, that means eight ICAP. So let's say this is the velocity of the man, okay? The wind seems to come to him from north, right? With certain velocity, right? So velocity of wind with respect to man is what I can claim it to be coming from the north. So let us call it as minus pj, poor Joe. Paragaditya, very good, okay? My question is what is the velocity of the wind? So Vm is what we are looking for. Oh, sorry, Vw is what we are looking for, okay? I'm sure all of you would have done this particular concept in physics that the velocity of any object with respect to another, let's say velocity of wind with respect to man is velocity of wind, actual velocity of wind minus actual velocity of man, okay? This is what we call as a relative velocity concept, okay? So now in the given question, or in the given scenario, let us say that the velocity of wind with respect, the velocity of wind in absolute sense, or you can say with respect to ground, all right? That would be wrong, okay? Is xi plus yj? So minus Vm is equal to eight i cap, correct? So from here, can I say eight i minus pj is equal to xi plus yj? If you compare i and j with each other, by the way, there is something very deep hidden inside it. When you compare x with eight and y with minus p, now tell you, why can you compare like that? Sir, because another contradiction that might have arisen is how do you multiply it? Like either they can be the same or they can be multiplied with the same ratio. But that won't be the case because you're adding vectors properly. No, it is basically like saying x plus y is equal to two plus three, so x is two and y is three. Is it true? ij is also there. Huh, but I mean, this is analogous to it. Like vectors. They are like vectors, okay? There's something which is called linearly independent, okay? Later on, we'll learn this and not in today's class for sure. I and j vectors are basically non-colinear and hence linearly independent. That means none of the vectors has got any component in the direction of the other. So they don't talk to each other. Yes. So when you are, later on, you'll study this. So when you are basically comparing or when you have, let's say, a n, b, s two vectors which are linearly independent and you are basically equating it to null vector, it can only happen when x is null, y is, sorry, x is zero, y is zero, if n, b are linearly independent, okay? This is a concept which we'll talk about later on, not now. As of now, okay, you can assume that when such a thing arises, you can compare i coefficient with i coefficient here and you can compare j coefficient with j coefficient. That is fine. Absolutely fine as of now. So one thing that you got from this was your x value, that is eight. But my job is not done, I need to get my y as well. But hold on, I have not used the second condition given to me. On doubling the speed, he realizes that the wind appears to come from northeast. Now, this guy starts walking at double the speed, 69. Okay? And then he realizes the wind is coming from northeast. The velocity of wind with respect to man is coming from northeast. Now, northeast I can take any vector to be, let's say, minus qi plus j kind of a thing, okay? So velocity of wind with respect to man is velocity of wind minus velocity, why am I writing m for w? Okay? Velocity of wind is still xi plus yj. Minus 16i, that is velocity of man. And this is given to us as minus qi plus, or sorry, minus qi minus qj, okay? That means minus q is equal to x minus 16, that means q is equal to eight, right? Because your x was eight. And minus q is equal to minus 16, or sorry, minus q is equal to plus y. That means y is minus eight. This is what I was looking for. So x is eight, y is minus eight. So the velocity of the wind was this. In other words, you can think as if the wind was coming from, the wind was coming from northwest. Isn't it? Yes, sir. So the wind was actually coming from northwest. Okay? So we have a lot of things to talk about. Of course, this is just the introduction class for us about vectors, okay? The main story will start when I start doing dot product, cross product, STP, VTP, vector equations. Then things will start becoming more heated up. Okay? And you probably feel the heat of the topic. Fine, so I'll stop over here because I don't want to give you any problem also because it'll take away at least six to seven minutes. So thank you so much.