 So people are still joining in though, but we will start discussion of what we are going to do today. So as you know and aware about that, we are done with the motion in 1D chapter. So in motion 1D, we have learned many aspects of the motion. But whatever we have learned, everything was happening in a straight line. The velocity, displacement, acceleration, everything was in one straight line. So now we are shifting our focus on motions which are not actually in a straight line. So if motion doesn't happen in a straight line, then how you will deal with it, that is what we are going to learn next. Fine. Now who will have done the last homework? Type in assignment, who are completed with the assignments? Just I want to check before I proceed. Are you guys done? Which one? The previous class, there was no assignment. Last class, there were two assignments. So can I do one thing like after today's class, I can send the assignment on your WhatsApp group for calculus on motion in 1D, use of calculus in the motion in 1D and do that. So I'll send it across. I'll send two assignments. Don't start doing the second one before doing the first one because the second one may be one level up. Now how many of you in your schools, UTs have already started or just about to start going on? Which school Anika? R&R, Sadhanain Neffel starting on Monday, begin this Monday. So I have heard that some of you got your marks already like in NPSHSR is where I have heard you guys have already got the marks. Do you mind telling me the marks which you got? Physics marks you got in NPSHSR. How much you got? Other section got it. And nowhere else you got your marks, no other school. All right, so probably I'll ask that in the next class, all right, next week. Anyways, so looks like we have good number of people joined in. All right, okay. Okay, fine. So we are going to start the next chapter, motion in a plane. Motion in a plane is about analyzing the movement, analyzing the movement of an object in a plane, right? So the movement can happen in a straight line, right? It goes straight like this or it can take a curved path, right? For example, if I just draw a path like this, right? If I draw a path like this, is this motion in 1D? What do you think? Is this motion in 1D when I draw like this? This is not motion in 1D, right? This is not motion in a straight line. Motion in 1D, motion in a straight line, same thing. So what we can call this is now do you all agree that no matter how I draw the line, no matter however it is, whatever line I draw, it will be in the plane of the screen only. Screen is a plane. On that plane only, every line will lie, whether it is a curved line or straight line, it will lie in this plane only. All of you agree, right? So motion in 2D is about movement of an object which is taking a curved path. Getting it? It is not a straight line path. It's a curved path, fine? It is a study of the curved path, which is lying in a single plane, all right? Now we, all of us see the movement of various things around us. So can you recognize few of the situation in which an object while moving can take a curved path? Which you see around yourself? Anyone? When a movement, when an object is moving, it can take a curved path. Any example? Good. When you kick a football, what will happen? It will take a curved path like this, right? When you drop a basketball, it goes like this, goes inside the basket and drops off, right? Many such thing. And what about the circular motion? Is circular motion the best example of the curved motion or not? What do you think? Right? So circular motion is also the motion in a plane. So in this chapter, in this chapter, we are going to take these few well-known examples and understand how to analyze the movement in a plane, okay? So I hope that sets the context properly that why we are studying this chapter, all right? This chapter will take lesser time compared to motion in 1D. Because in motion in one dimension, we also got introduced to what is displacement, what is velocity, what is acceleration and things like that, right? So whatever introduction of velocity, acceleration and everything was there in motion in 1D, same thing is true here also. So nothing is going to change, right? We are just going to talk about few examples here, all right? Now comes the main issue. When we study things that are moving in a plane, all right? Then the problem is that we cannot study it in a very simplified manner. What do I mean by simplified manner? For example, when let's say velocity was going like this and acceleration was in opposite direction, right? How we took care of the directions? Velocity, if I take positive, acceleration becomes what? If velocity is positive, acceleration is? How I take care of the directions? Do you all remember we used sign for the directions? All of you remember or not? Positive sign, negative sign, we use those symbols for the directions, all right? But what if velocity is this way but acceleration is making 30 degree with it? Can I use the sign to take care of the directions? Can I do that? If velocity is this way, acceleration makes 30 degree to the velocity. I cannot use the plus or minus sign for the directions. Are you getting it? So using plus or minus sign to take care of the direction is an oversimplified version of use of vectors. Which doesn't work when the velocity and acceleration, they are not in one straight line. So we cannot study motion in a plane by using oversimplified version of vector. We need to actually learn how vector works, all right? And that is what we will do in today's session. So today's session, mostly we are going to talk about the mathematical part of the mathematical tool which is called vectors, which will help us analyze all the movements. So because again, we cannot use plus or minus sign in motion in 2D to take care of the direction. We have to use vectors directly, okay? So, all right, somebody is asking, can you give an example where velocity and acceleration are not in the same direction? All right? Can anybody answer this question where velocity and acceleration are not in the same direction? There is some angle between them. Anyone wants to answer? Okay. Correct. Others, anybody else? Right, many things are there, right? So for example, you throw a ball. Ball will take this kind of path. Ball goes up and down like this. This is the velocity direction in which the ball is moving. But the only force is gravity, which is creating acceleration vertically downwards. Acceleration is down and velocity keeps changing its direction. Does it clear, Sadhana? Velocity keeps changing its direction. But only force applied is gravity and mg is equal to mass and acceleration. So, acceleration is g in the direction of the gravity force. So, acceleration is always vertically down. All right? Is there anyone here from NAFEL attending the today's class? But I think you have the chemistry class, right? In NPS Rajay Jainagar. You have a chemistry class going on there. Those who are from NAFEL right now attending here, the classes. But anyways. No, no, the reminder was sent. You recently joined Sadhana, is it? When did you join? When did you join? When did you start joining? As in one or two days before only? Okay, no problem. Are you attending this session anyways? Because you have not done this one also. So, after the class gets over, please get in touch with me. Okay, clear? All right, everyone, let us get going with vectors. Last session, it was heavily loaded with the calculus. Today's session, it is loaded with vectors. Okay? So, after today's session, I may forget that I am a physics teacher, only teaching mathematics for two classes now. How many of you were there, by the way, in the bridge program? When I was teaching vectors, how many of you were there in the bridge program? Many of you were there. Oh, all right. So, those who have already attended the vectors with me, probably it is a good revision. All right. Some of you have already watched the recordings also. Not a problem. Let us see if I can find something new for you. Anyways, so, all of you, you know, in units and measurement chapter, we discussed, what we discussed that in physics, there are various quantities. There are so many quantities. And for every quantity, we measure differently. We measure mass differently. We measure length differently, time differently. Okay? So, there are different, different ways we measure different, different things. Fine. And these are what? These are quantities. Mass is a quantity. That is the reason why you have to put a number to it. You have to put a number that it is number 5, 5 kg it is. All right? So, likewise in physics, there are so many quantities. You can keep on listing down. For example, let me list down couple of them. You can also help me. Mass is there. Length is there. Time. Everyone. Force. Acceleration. Velocity. Temperature. Current. Energy. Anything else? Displacement. Momentum. Distance. Okay? Flux. What else? Charge. Current is written. Current is written already. Anything else? Anything else comes into mind? That's it. There are so many actually. There is angle. Angular velocity. Angular momentum. There are so many others. Okay, so let's not be comprehensive here. Just that I am telling you that there are so many things in physics for which you need to put a number to it. Volume is also one. Okay? So there are few things. If you just tell a number that is sufficient. Let's say mass. Length. Time. If I tell you the time is 2 seconds. That is good enough. Entire information is conveyed. Alright? The mass is 5 kg. Sufficient. Nothing else you have to tell. Alright? But if I tell you velocity is 5 m per second. Is that sufficient information? What do you think? Will that help you to analyze where the object is? After some time. If I tell you velocity is something. Consent velocity only. Consent velocity. I tell you the velocity is 2 m per second. After 2 seconds where the object will be? It will say 2 into 2 4 meters. But I am asking you where exactly it will be? You cannot tell. Why? Because I am not telling you. I am not telling you which direction that object is moving. Okay? It is like you have started from your home let us say and you are going with a speed of 10 m per second. You are going with 10 m per second and I am asking you how much time it will take for you to reach the school. What if you are not going towards the school? You are probably going to a mall. Right? In the opposite direction you are moving. So I cannot find out until you tell me that I am moving in that direction. Right? If I tell you, let us talk about the displacement now. If I tell you that there is a treasure 2 km from where you are sitting. Will you be able to find out? Will you be able to find out if I just tell you it is at 2 km distance away from where you are sitting? Is it sufficient information for you to find out where the treasure is? What do you think? Is it not sufficient? You need to know exactly which direction it is in. Right? So there are many quantities in physics. There are many quantities in physics for which just telling the magnitude is not sufficient. Okay? Just telling a number is not sufficient. We have to also find out the direction of it. Alright? So when I am quantifying something. Right? When I am putting some number against a physics quantity many times I have to tell that direction also. Right? Many times somebody might be asking directions from you when that person might be going somewhere from one place to the other place. How you tell? You tell that go straight, take second left then go for 2 km forward, take the right turn after that. Like that you tell the directions, right? So many times what happens is we try to tell the direction by using words. We try to explain the direction. Okay? We are explaining the direction by telling that okay you have to take a right turn then take a left turn and things like that. But still we are not quantifying the direction. It is like you are telling okay mass is very heavy time is very large force is very less that kind of thing we are doing when we are explaining the direction by using English. So we need to quantify it by using some mathematics so that we can analyze it further. Okay? Otherwise we will just know the direction. What will happen after you know the direction? Will you be able to do the analysis of that situation? That is possible only when you mathematically put some value to it. Okay? And that's the reason why we are learning vectors. Okay? So vectors is as big a universe as big the scalars are. For let me go to this. What I am telling you is that till now our entire focus was on scalars. Okay? We only focused on the scalar quantities mass, length, time, charge and all that. Fine? Since our childhood grade one or two onwards we were learning about the numbers how to I mean how you write number one, how you write two then how to add the two numbers how to multiply the two numbers how to divide the two numbers everything that we have learned till now was related to scalars. Numbers. Alright? Now I am telling you that there is another universe that exists. This is one universe. This is another universe. Okay? This is a scalar and this is vector. They are different from each other. Alright? So just like as a kid when we were learning about the scalars we never assumed anything. We learned everything from scratch how you add how you multiply and stuff like that. Same way exactly same way. We need to learn about the vectors. We need to learn what is a vector, how to write the vector, how to add the vectors, how to multiply the vectors. Right? So whatever we have learned for the scalars may not be directly used for the vectors. So never try to utilize whatever you have learned in scalars. Don't try to think that everything that you learn in scalars is directly applicable to the vectors. So you need to learn it from the scratch. How to add, multiply and stuff like that. So that is the reason why in today's session we are going to learn about these vectors from scratch as if we are a kid. Alright? So that is something which you should keep in your mind. Alright? Now here you can see some of the quantities which are scalar quantities, mass, volume, distance, all of these they do not require any sense of direction. So you guys are making your notes or not? And when I am showing something on here, you guys make notes or not? Okay. So you don't need to write this entire thing but at least few of these things better you write down. Okay, few of these things you write down. Alright? Now let us move forward. So you need to behave or you need to treat as if you don't know anything about the vectors. Then only you will learn the most. Okay? Now vector is a quantity which should tell you the direction. Okay? Shouldn't it also tell the magnitude? What do you think? If I ask you about the velocity, will you just tell the direction? No. Will you just tell the magnitude? No. You have to tell both. Right? You have to tell both. So it will have both direction as well as the magnitude. Okay? Now since it has both direction and magnitude, there should be a way to represent it. A way to represent it. Alright? So since it has a sense of direction, we can represent it geometrically also. Geometrically we can represent in one way and algebraically we can represent it in another way. Okay? Broadly what I am trying to say is what I am trying to say is that vectors broadly the analysis of the vectors can be divided into two parts. Geometric analysis and algebraic analysis. Okay? I think algebraic spelling is incorrect, right? There is a spelling mistake. Can anybody help? What is this? Oh, is it only me? It doesn't know. Oh. Algebraic. Okay. Alright. So vectors you can see that broadly divided into two parts. Geometrical analysis and algebraic analysis. Alright? So these things are slightly disjoint from each other. So we are going to study about the geometrical analysis first. So before studying the geometrical analysis we need to first understand how you can geometrically represent a vector. Alright? You can represent in any which way which is making sense. Right? You can represent it like this. You can say that the tip represents the direction of the vector and the area of the strangle represents the magnitude of the vector. But this will have its own complications. Alright? So what is the easiest way to represent a vector? Can anybody highlight? Good. So you can represent a vector by using an arrow. This is an arrow. Alright? An arrow. Why? Because arrow is the best way to tell the direction by using the head and the length of the arrow can represent the magnitude. So geometrically vectors it is not that God has come from the sky and told us that this is how the vector should be. It is a human creation. We have created it. So you can assume anything. So humans when they create, they look for the comfort and the ease. So that's the reason why we all have agreed that easily you can represent a vector by using an arrow. So we cannot debate upon that why arrow is used, why not this, why not that. You can use anything that makes sense. Alright? So when we represent it like an arrow, the length represented as the magnitude and the head represented by the direction. Okay? This keep in your mind. Clear? Anybody has any doubt? Any doubt anybody has? No doubts? Now tell me if I move this vector, I just duplicate it. Now if I take this vector, move parallel to itself and I move it over there. I have moved this vector over there. Has the direction changed? When I move this vector over there, has the direction changed? The direction is still on the right hand side. The direction is same. Has the length changed? Has the length changed? Length also is not changed. So my dear friends, we can move the vector parallel to itself without affecting its value. So write down a vector, vector can be moved parallel to itself without affecting its value. Okay? It doesn't get affected. It remains the same. Got it? I hope this is... Now is this something you need to vectors only? What do you think? Does this... Is this something which is completely new? Is it? It's not something completely new. For example, let's say you write number 3 here. You move this number 3 from here to there. It still remains 3. Okay? It still remains 3. In fact, if you rotate this 3 little bit, does it still remain 3 or not? Even if you have rotated, does it remain 3? It remains 3 only. Okay? That doesn't change. Then if you rotate this vector a bit like that, length is same. Has the vector changed? Has the vector changed? Right? Vector has changed because the direction is changed. Direction is an integral part of the vectors. Alright? So direction is changed, vector is changed. So this is not something unique only to the vector. And in fact, you cannot rotate the vector, but you can move it parallel to itself. I hope these things are very, very clear. Okay? Now, always keep in mind we are talking about the geometrical analysis, algebraic analysis. We will do later. We will not do it right now. We are just focusing on the geometrical part of it. Alright? Remember that. Now, when it comes to a variable, which is a scalar variable, we use letters like X, Y, Z, all of us remember or not? In equation like 3X plus 2Y is equal to 6 like that, X, Y, and Z they are variables, right? They are, they are what variables? They are scalar variables. They are scalar variables. Similarly, a variable in a vector can also be used, can also be represented by letters only, but when you write a letter, put a bar also on top of it like this. When you write like that, put a bar at the top, it means that you are talking about a vector variable. Alright? Many books, they will represent it like that also. It is same. It is one and the same thing. Alright? So these are the few things that are assumed. Okay? Every time any story starts with certain assumption, right? Assume that there is a king which is ruling the country at this time and then that is how the story starts. Similarly, the story of vector starts from here. All of the assumptions that we have made is here only. Okay? Now we can talk about using vector to analyze something or how you add, how you multiply and stuff like that. Anybody has any doubts till now? Type in quick. No doubts. Okay? Since there are no doubts, I will move forward. So remember, whatever I am scribbling, that will be erased. Alright? So you need to keep writing that at least. So, same thing which we were talking about. Here, vector quantities are the quantities which have direction as well as magnitude. Now, this is just an explanation. When you add this to it, it becomes definition. This is also an important part of any vector. A vector must follow vector law of addition and subtraction. Okay? Now, what it is, what it is, we are going to discuss a little later, but it is a law which every vector must follow. Okay? I will just give you a brief of it. For example, let's say, let's say you have this a box. You apply it 2 Newton from this side and 2 Newton from the opposite side. Total force is how much? Everyone? What is the total force here? 0, why 0? Why not 4? Why not 4? Because you are considering the direction also? Yes or no? Because you are considering the direction also? All of us agree to that? Right? Because you are considering the direction, that is why we are saying it is 0. Now, imagine like this 2 ampere current coming from here 2 ampere current coming from there. How much current will go there? What current will go there? Everyone? There are 4 ampere. 4 ampere. So, current has current has the direction and magnitude. Right? It is coming from the left coming from the right. It has a direction as well as the magnitude. But it doesn't behave like a vector when you add it. Right? That is the reason why current despite having magnitude and direction is not a vector quantity. All right? So, is it a scalar? You can call it a tensor quantity. Okay? But then yes, for the for our purpose of grade 11th and 12th you can treat it like a scalar, but you have to take care of the direction. Right? You need it is not a completely a scalar quantity. Right? But it Sir, because NCRT had a question which was like which of the following are vectors of scalar quantities and in that current was given as the right answer. That's why I was asking. Current was given as a scalar quantity. Right? Right. So, then you that is what I am telling you. You need to mark it as a scalar for your school purposes. But then when you are analyzing something very complex circuit you need to know the direction or not or just right? You need to know it from which direction current is coming in. So that treatment of the vector is little different than all the scalar quantities. Okay? But then if it comes in your school exam you write it like a scalar. Okay? Now this is what we have already discussed that a vector will have a head it will have a tail the magnitude will be the length and direction is represented by the head. Okay? This we already discussed. Now these are the some things these are something that you need to copy down. Okay? So everyone copy down there is something called a unit vector as the name suggests as the name suggests unit vector will we have this for the test Sunday test you are talking about Sunday's test Sunday's test probably not but the test after that which will be next Sunday then it will come every Sunday you have test on every Sunday Okay? Surprise, surprise you will be having you have the four pattern test right? You have a CT test you have a J main test J advanced and need test right? If you want to write everything need CT J main J advanced then every Sunday there is a test because four weeks there will be a test but if you want to only focus on CT then only once in a month the test will happen understood? Alright unit vector as a name suggests it has only the direction but magnitude is one alright? magnitude is only one now tell me now tell me have you ever you know defined something as a one unit and that was very useful has it ever happened? is it useful to define one unit of anything? like? correct like mass like mass we defined one kg right? once you define what is one then you can find out what is two what is three what is 2.5 what is 3.5 times is it clear? so that is the reason why defining what is a unit vector is important although it is very easy but still we specify it alright? now there is something called as null vector alright? null vector is an analogous to zero for the scalar fine so like I told you the scalar the universe is different the vector the universe is different so we cannot say that nothing of a vector is zero because when you say zero it implies that you are talking about a scalar quantity although it is zero only but we name it as null vector alright? null vector is a vector having zero magnitude and no direction just like zero of a scalar okay? then equal vectors when I say this vector is equal to that vector they are not just equal in the magnitude they are also equal in the direction alright? just like if this is one vector this is the other vector they can never be equal although their magnitudes may be equal but their directions are different they can never be equal alright? there can be vectors which we can call it a parallel vectors two vectors like this they are parallel to each other different magnitude but parallel okay? now these last two things are not as important these are the obvious things but defining unit vector and null vector is important somebody is saying parallel vectors are equal vectors is it correct? what do you think? others are parallel vectors are equal vectors? they need not be right? they can be of different different lengths one is just only if they are of same magnitude correct? good so if two vectors having the same magnitude same direction that means they are parallel to each other then only they are equal alright? let me go to the next okay these are like dictionary of vectors it is good to know these things beforehand itself anti-parallel what do you think? parallel vectors are parallel vectors they are in the same direction so anti of anything is in opposite direction they are in opposite direction completely opposite so one vector is in right hand side let's say sorry left hand side the other vector will be in which direction? it will be in that direction so these two vectors are in opposite directions alright? now does it need to be equal magnitude? need not be just about the direction when I say when I say one vector is negative of the other we have already learned in the motion one d if velocity vector this way efficient vector that way then one is negative of the other right? we take out the direction with the negative sign alright? okay somebody is asking why both negative and anti-parallel they mean the same right? no when I say negative vector when I say A is equal to negative of B what does it imply? it implies that vector A this way and vector B that way and not only just about the directions their magnitudes are also equal got it Pratik? this is A and this will be minus B not minus B this is B only okay A is equal to negative of B alright? anti-parallel just means one is in opposite direction alright? concurrent coplanar orthogonal if I hide the definition let's say if I don't show you these things right? what do you think concurrent should be? what do you think concurrent should be? everyone you can speak up also concurrent means starting from a point both the vectors starting from the same point coplanar means what? what do you think coplanar meaning? they are in the same plane vectors are in the same plane okay they start from the same point concurrent orthogonal what do you think? more than two vectors also be in the same plane correct? all the vectors are coplanar then however number of vectors are there orthogonal means what? orthogonal means that one vector is coplanar of other coplanar to other okay? this is how you can say so these are the obvious definitions alright? these are the obvious things should come out naturally alright? don't try to mug it up it is of no use mugging up these things alright? just for your information's sake what is orthogonal? orthogonal means one vector is perpendicular to the other ortho means perpendicular orthogonal means have you guys heard of orthogonal word before? have you guys heard of orthogonal word before? no okay orthogonal means 90 degrees alright this is something we already discussed RNA saying he knows only orthopedic orthopedic orthopedic is related to the bones that is different looks like a medical term yeah yeah yeah but I thought orthogonal is something that you guys have heard already but anyways so this we already discussed that vector there can be two kinds of analysis graphical as well as algebraic so nothing to note down here this we already know of this is the way we represent a vector okay somebody is saying if two vectors are such that head of one vector touches the tail of the other and this forms 90 degree is it still orthogonal? what do you think everyone? if two vector touch each other and they are 90 degree to each other like that are they orthogonal or not? like this two vectors are they are they orthogonal? they are orthogonal they are orthogonal alright just that 90 degree should be there right now I am going to basically go beyond the slides little bit because slides are prepared long back I want to modify few things here okay so the when I say A with a bar like this it represents a vector variable all of us agree it represents a vector variable alright and and the vector variable A could be this this could be the vector A fine this is how we represent a vector A alright and like we have just discussed that if one vector is like this other vector is in opposite direction let us say two forces are there okay two forces are applied which are equal and opposite A and B they are equal and opposite we know that net force will be zero two Newton from the left hand side two Newton from the right hand side when you add it up it should become zero right so we already know that some of the two vectors should give us zero so that is the reason why A should be equal to negative of B alright so basically what does it mean if this is A if I reverse the direction what it will become everyone if this is A this is what correct minus A alright so I can just multiply negative one with the vector and the vector changes its direction just reverses the direction alright okay that is point number one second is this vector variable when I write like this it has both magnitude and direction inside the same variable alright suppose I want to I want to extract the magnitude out of it I want to just find out the magnitude the way we represent magnitude of the vector is like this it is represented like that just the way of representation alright so if you anywhere if you see this kind of symbol it implies that it implies what it implies that I am talking about only the magnitude of a vector fine now tell me can a magnitude of a vector be negative what do you think can the magnitude of a vector be a negative quantity alright some of you are saying yes some of you are saying no now tell me tell me the magnitude of a vector if I draw a line if I draw an arrow magnitude is the length of an arrow magnitude of vector is the length of the arrow can the length of the arrow be negative ever it cannot be right so the magnitude of the vector can never be a negative quantity okay it can never be a negative quantity understand that fine this is how we represent the magnitude now suppose suppose I want to find out a unit vector in the direction of A how you will find out we will divide vector A with its own magnitude like that this is a unit vector when you put A with a cap when you put like this on top of A it means that you are talking about a unit vector alright so mod of a unit vector is mod of this entire thing mod of mod of A is mod A only so mod A by mod A which is 1 so the magnitude of the unit vector is 1 and how it is done you divide vector A by its own magnitude is it clear to all of you type in okay I hope no doubts still no doubts okay now one more thing one more thing this is about the angle right down angle between the two vectors how do we define and find out the angle between the two vectors that is what we are going to discuss fine so let us say we have a vector this this is vector A and let us say we have another vector like this vector B okay yes costume don't worry I will share the slides with you and I will share the video also you don't need to worry about it because previous slide I cannot go now everything will be erased fine now tell me just leave little bit of space and move forward okay now tell me what is the angle between A and B as far as this example concern is the angle between A and B this theta or phi is the angle between A and B okay some of you saying theta some of you saying phi okay now now tell me tell me how much how much A should be rotated to B in the direction of B can you answer this should it rotate by theta or phi phi phi okay now answer to me how much all of you should write with me okay when I am writing you also should write how much B to be rotated to B in the direction of A theta or phi what I can do is that I can move a vector parallel to itself so I can move this B like this it still remains B only isn't it it will remain B only it will not change it will be phi only or not how much B should be rotated to B in the direction of A this vector and this vector are same I have moved the vector parallel to itself it will be phi only yes or no type in do you all understand B rotates by phi to be in the direction of A and A can rotate by phi to be in the direction of B all of us agree everyone type in type in quick all of you why not theta somebody asking why not theta now tell me everyone tell me if let's say this is A this is A if I rotate by theta it will be in the direction of B or in opposite direction of B what do you think it will be in the opposite direction so that is why not theta alright now tell me which one will be the logical choice of angle between the 2 vectors theta or phi it will be phi only right it will be phi so when we try to find out the angle between the 2 vectors the first step to find out the angle between the 2 vector is connect the 2 vectors connect the 2 vectors tail to tail tail to tail then measure the angle, this angle. Now how you suppose vector is not connected to a tail to tail initially then what you have to do? What do you have to do? If it is not connected to a tail to tail already then what we can do to make sure that they are connected tail to tail. I can remember we can displace or move a vector parallel to itself without affecting its value. Remember that, always remember that this is a common thing that you will be doing every time. Has anyone has any doubts till now? Type in, no doubts. Some of you might find it repetitive, you might have learnt the vectors already but I am trying to go to the deep inside like the core of it, why it is, how it is. I am not just telling you that as you angle is theta. We are discussing about why it is theta, why it is phi and stuff like that. Now once we learn, till now what we have learned? We have learned that there is something called vector exists. We have learned that we need it. We have learned that how you write the vectors, how you represent the vector. Now once you have done, just a second somebody is asking a doubt is asking can we extend the vectors? Are you saying that can you increase the length of the vector? That is what you are saying? Now when you extend the vector, will the vector remain the same vector or not? Or will it change? It changes, magnitude changes, it does not remain the same. Okay, green tea is enough. Fine, green tea. Alright, it does not remain the same. For example, 2 Newton is applied like this in this direction 2 Newton force. You are telling me that you want to change the length, you want to make it 4 Newton. Is 4 Newton in the same direction different from 2 Newton or not? It is different. It cannot be same. You cannot extend it. You can move it. Okay, take the vector from here and move it over there but you cannot increase the length. Alright, at the same time, at the same time when let us say you are talking about a huge vector, let us say 5 kilometer displacement you have to show. 5 kilometer displacement. Will you draw 5 kilometer long vector? Does it make sense? 5 kilometer long vector? It does not make sense. You will draw a 5 centimeter long vector. You will say that 1 centimeter is 1 kilometer. So, you can scale it down. Alright, but everything over there should be scaling down. Okay. Alright, let us. So, what I was talking about, we have learned everything about how we can represent the vector, what is the angle of the two vectors and what is unit vector, how we write it and stuff like that. The next logical step. You remember when we learned about the numbers, you were very, very little kids grade 1, grade 2 where I see my daughter, she is in grade 3 right from the nursery and all. She learned about the numbers first that 1, 2, 3, 4 like that she remembered like that. That is what we did just now. After that in grade 2 onwards, she started learning how you can add the numbers and that is what we are going to do now. How to add the two vectors. Then in grade 3, she learned, she is learning about how you multiply the two numbers. So, we are going to see after learning how to add the vectors, we will see how to multiply with the vectors. Okay. So, next thing is how we can add the two vectors. Don't worry about these slides right now. Now, I am going to debate first with you that why when we add the two vector, it makes sense that this is the sum of the two vectors. Let us talk about that. Suppose there is a vector like this. This is a vector. Vector A. Okay. There is another vector like this. This is vector B. Alright. What is there that this pink vector that you see here, this one, this is actually the sum of the two vectors. This is the sum of the two vectors. Now, why it is a sum of the two vectors that we will understand by taking an example, by taking displacement as an example. Okay. So, can anyone try to explain why this makes sense to have some of the two vectors? Everyone? You can just type in that yes, I want to speak, then I will tell you that yes, you can speak. Otherwise, you will start speaking. Who want to explain that A plus B, this one makes sense to have some of the two vectors. Alright. No one. No one. Okay. Unmute and explain. So, if you start from the starting point of A and move to here. Yes. You move till the end and then you change the direction and then go to the end point of B. So, the displacement will be the distance from starting to ending point. So, which is between 1 and 3. Hence, the sum is that not. Okay. Do you all understand this? He is saying that first, the displacement is 1 to 2. Second displacement is 2 to 3. So, the sum of the displacement should be from initial to the final point. So, that is why this 1 to 3, this vector makes sense to be the sum of the two vectors. Is it clear to everyone? Type in. Now, tell me whenever you add the two vectors, whenever you add the two vector, will you get a triangle every time? Will you get a triangle every time? As some are saying, may not. Tell me when it is not a triangle. When you will not get a triangle. Probably you mean to say like this. This is what you mean? Collinear? Like this? Yes or no? This is what you are saying? Right? What if I tell you that it is a triangle? It is a triangle and one of the angles of the triangle is 180 degree. Makes sense or not? The other two angles are 0 and the one angle is 180 degree. So, technically it is a triangle, but I know that it is, I mean you do not see three sides separately. Alright? Okay, hard to imagine, just imagine this. This is a triangle? Is it a triangle or not? It is a triangle. Now, this angle is very small. How small you can make these two angles so that it still remains a triangle? What do you think? How small these two angles you can make so that it is still remaining the triangle? Can you answer that question? You cannot answer? Okay, you are saying exactly zero. When you say exactly zero, it becomes a straight line, right? Otherwise, you can say 0.001 degrees. I will say no, 0.0001 degrees. So, there is no end to it, right? There is no end to it. So, I am saying that when these two angles, they tend to zero, you get this only. So, everything is a triangle only. Technically, everything is a triangle. Now, you are able to imagine, Aryan. Okay. Now, since when we add the two vectors, every time we get a triangle, what do you think is a good name for this law, the way we add the two vectors? What should be the law? Good. So, it should be triangle law of addition, triangle law of addition. And every vector, every vector must follow this law. Okay. If you find something having a direction and that something doesn't follow triangle law of addition, then that something is not a vector. It is not a vector. For example, all right. So, this is a triangle law of addition. I hope it makes sense. Do you want to do certain questions on it? Certain questions you want to do it? Let's take one or two questions on it. Everyone, draw this diagram in your notebook. Last year, there was a student who used to take screenshots and save in a folder as notes. We found out after a month or so, after the test happened, we found out that why this guy is not getting any marks in the test. And then we got to know ultimately that this guy was never making notes, was taking screenshots. Okay. So, don't do like that in case any one of you are doing it. Anyways, these are the two vectors A and B. Try to add these two vectors and let me know once you're done. I want to find out A plus B, what it is. Done. Pandey is done. No, no, don't tell me how to do it. You just type in that you're done. Okay. Is there anyone needing more time? Anybody else needing more time? All right, done. Okay. So, now before adding any two vectors, what is the first step? The first step is you need to connect. Write down connect the two vectors head to tail. To connect them head to tail. All right. And then create a triangle. Once a triangle is created, the third side is the sum of the two vectors. All right. Can we connect head to head? No, you cannot connect head to head. All right. Then you can't do it like that. The reason it is simple. We have explained like this only, right? The particle is going from 1 to 2, then 2 to 3. So, this is a total displacement. That's how triangle is made. All right. If you connect, let's say head to head like this, will you be able to say that it went from 1 to 2, then 2 to 3, you can't, you will not be able to see the direction of the second vector is like this. You can't imagine like that. Is it clear or not? But it went from 1 to 2 and 3 to 2. So, you're not adding it up. Two displacements should happen one after the other, right? You went from 1 to 2 and suddenly you are at 3. You can't suddenly go to 3, right? You have to travel from 2 onwards to the next place. All right. Now, what I'll do is that I will move B parallel to itself and I connect like this. Once I connect it, it remains B only or not or the vector changes. What do you think? It remains B, right? A vector can be moved parallel to itself and then this third side, my dear friends, is the sum of the two vectors. This one. All right. This is A plus B. All of you, is it very clear? Quickly type in. Quickly type in. All right. Sadhana, I'm coming to that. Hold on. Now, if you call it A plus B, then what is B plus A? Why same? Zayan is saying same. Why same? Because you are correlating with scalars. You're saying 2 plus 3 is equal to 3 plus 2. So, same thing should work with vectors also. Are you comparing with this kind of setup? Yes or no? Comparing with this kind of setup. So, what I told you, what I told you, these are the two different universes. You cannot compare what happens with the numbers. Same thing. There is no guarantee that will happen with the vectors. Now, if you move B parallel to itself, if you're saying that is A plus B, then if you move A parallel to itself, what you can say that as rather than moving B, you're moving A now parallel to itself. You can move it like this. Now, it is connecting head to tail. Is it no? Now, it is connecting head to tail. So, you will get this as A plus B plus A now. This is B plus A. And as it turns out, as it turns out that these two blue lines, they are parallel. They are parallel and they are having the same length. So, that is the reason why A plus B is equal to B plus A. Remember this. Clear? This is the proof of A plus B is equal to B plus A. Okay. Yes, they are equal. In fact, I could have moved A over there. I could have moved A like this. This is A only. This is A. And A and B are connected head to tail or not? A or B are connected head to tail? All of you agree or not? So, you get a parallelogram. You get a parallelogram. So, parallelogram, this is A plus B and this is also B plus A. Like that also you can say. Got it? I hope things are making sense. Anybody has any doubts? Shall we take more examples? Let's take more examples. This is, let's say, vector A. Okay. And this one is vector B. You have to find out what is A minus B. Try to draw this. You don't need to be perfectly right in terms of diagram. Whatever something like this you draw in your notebook and find out A minus B. Let me know once you're done. Many of us are done. So, if B is like that, where is minus B? All of us agree that A minus B is nothing but A minus B is nothing but A plus minus B. I am adding A and minus B. All right? So, if we reverse B like this, there will be a minus B. All of us are agreeing. There is minus B. So, this is what? This will be A minus B. Okay? Clear? I hope these things are making sense. Now, tell me where is B minus A? How will we join minus A and minus B? Where is minus A? It is A and minus B. It is A and minus B, right? Can we join? Yeah, why not? But if you do it like this, minus A minus B is what you will get if you join minus A and minus B. I want A minus B. You are getting minus A minus B. B minus A. Tell me once you're done. Okay. So, what we do is that A is this. So, I will connect minus A with B. This is minus A. B and minus A, I have connected head to tail. So, B minus A is B plus minus A. All right? This vector, this is B minus A. Now, these two lines B minus A and A minus B, are they looking parallel or not? They are looking parallel, but the directions are opposite. You can see clearly. So, A minus B is equal to minus of B minus A, obviously. Right? So, negative of this vector is that vector. They are parallel and equal magnitude. Sorry, anti-parallel and equal magnitude. Clear or not? Everyone type in, sir, I connected minus A, it parallel and opposite to A. Will minus A parallel and opposite to A. I didn't get what you're saying. Can you unmute and speak? Sir, basically, I connected take took B as the origin, the point where A and B touches, and then I took that opposite to it as minus A. Yeah, from that, it's a basic. This is minus A. Yeah. No, no, no. The same thing towards the opposite side. That is correct, but now what will happen? This is minus A and this is B. Are they connected head to tail? No, that's why I took it to the opposite side, to the right hand side of B. Well, you did it like this. Yeah, yeah. That is correct only. You get the same vector only. Once you do that, see what he's saying it, you unmute yourself. Sorry, mute yourself. This is what he did if he's saying that this is minus A. So, B minus A would be this. There's nothing wrong with it. Now, this becomes a parallelogram. So, this side and this side parallel and same magnitude. That is also B minus A only. Is it clear RN? Not RN, Aditya. Yes, sir. Okay, the same thing you get. So, now one last thing before I progress to the next concept, tell me the magnitude of the sum of two vectors, the magnitude of the sum of the two vectors. Can it ever be more than the magnitude of some of the magnitude of the individual vectors? You understand my question? Great. Let me explain it again what I just asked. Yeah, repeat. Listen here. This is A, this is B. Where is A plus B? A plus B is this. All of us agree that this is A plus B. Length of A plus B is magnitude of A plus B. Agree or not? Which is A plus B? What I'm saying is, can this ever be greater than sum of magnitude of A and B? Can this happen? That is my question. If not, why? Okay, nobody has given a convincing answer yet. No one. Have you learned about the triangle inequality? Have you learned about a triangle inequality? The sum of the two sides will always be greater than the third side. Same thing is applicable here. Ultimately, magnitude of A plus B is what? Length of A plus B. Magnitude of A is what? Length of A, micro B is length of B and there's a triangle ultimately. Every time you get a triangle, so this can never happen. That the third, any of the side is greater than some of the other two sides. It can never happen. At max, it can become equal. When it becomes equal, when both other two sides are, I mean, it is technically a triangle, but actually everything is happening in a straight line. So A plus B will always be less than or equal to magnitude of A plus magnitude of B. This is only for the magnitude. This is something unique to a vector. Such kind of thing doesn't exist for numbers. So three plus two is five. Even if you take mod, without taking mod, not taking mod, it is always three plus two is equal to five. But that kind of unique thing happens with a vector. When you add the two vectors, whatever comes out, the length of that vector will always be less than or at max equal to the length, the sum of the length of the other two vectors. Remember this. Now, let us move progress to the next thing about the vectors. The next thing about the vector is the parallelogram law of addition. But let me just browse through the slides so that we sure that we are not missing anything. We have learned this already. You don't need to note it down. Don't worry. We have already done this just now. We have done this also, reversing the vector to find out A minus B. This also we did. This also we did. How to find the angle between the two vectors. So now tell me, when I write down something on the board and you learn that way better, or when I show that as a slide only, like this one, I don't need to write anything. I can just project it. That is better. Or when I write it down and explain it, that is better. Writing down. So what I'll do from next time onwards is I'll use slides only for the heading and diagrams maybe whatever is written that I will write myself. Clear? Now, I will come back to this after discussing the parallelogram law. Okay, then the written parts we shared. Yeah, why you want written parts to be shared? If I start sharing the written parts, you will not write anything. I know. Write down parallelogram law of addition. By the way, have you seen on the Learnist, there is already a written part of every class that we take? No, that was for last year. Sorry about that. We upload the slides on the there as well. What I'll do, I'll share all the written parts at once the chapter gets over. At the end, I'll upload all of it. Okay, that may be the last years. But still, I mean, the vector doesn't change right year on year. Whatever was the last year vectors, we did same thing. Just that maybe the flow will be slightly different. So that is fine. Right? Parallelogram law of vector addition. Now, this is nothing new. It is a triangle law only, but people are calling it parallelogram law and creating its importance. I do not know why. But since it is there, people are giving it importance, so we need to learn also. I mean, like I told you, nothing to be learned. It is same as triangle of addition. But in triangle of addition, how you connect the two vectors head to tail or tail to tail, head to head. How you connect? We connect head to tail, head to tail. Whereas in parallelogram law of addition, what we do is that we connect tail to tail only. Nothing special. I'll just show you. It's a very easy and straightforward thing. Two vectors are like that. Let us say these are two vectors. Okay, this vector A, this is vector B. Now, I want to add these two vectors. I do not know parallelogram law of addition. I want to use triangle law of addition. So then what you do? You take this vector B, move it parallel to itself and connect like this. Okay? When you move it parallel to itself, you know that this will be the A plus B. Yes or no? Type in. You could have moved A parallel to itself. You could have moved A parallel to itself, this one. Like this. Like this, you could have moved. When you move both A and B to find out A plus B, what get generated is, what get generated? Parallelogram. Parallelogram get generated. Okay? So using triangle law of addition only, you could have added. But when you move both A and B, parallelogram get generated. Right? So going forward, whenever two vectors are given, which are connected tail to tail, like this, these two vectors are given. Will you be able, will we be able to generate parallelogram out of any two vectors that are connected tail to tail? Yes or no? Right? So I can draw parallelogram. I can draw it. Whenever you have such situation, the diagonal that starts from the joining of the two vector is the sum of this vector and that vector. Okay? This is A. This is B. The diagonal is A plus B. Clear? Pretty simple. All of you and somebody is saying, when do you triangle law? When do you parallelogram law? I mean, both are same only. Do you all agree both are same? Everyone, do you agree both are same? Both are same. Now, if one diagonal is A plus B, can you tell me what is this other diagonal? This one is what? What is this diagonal? All of you. Yeah, aren't, don't worry. We are going to again discuss this. Hold on. What is this vector? This is B. This is A. This is B. Connected tail to tail. Connected tail to tail. All right? Now, if you reverse B, this will become what? This becomes what? If you reverse B, it will become minus B. Now, you can see that A and minus B are connected head to tail. So, this vector is A minus B. All right? So, when you create a parallelogram out of the two vectors that are connected tail to tail, one diagonal represents sum of the two vectors. The other one represents the difference of the two vectors. Clear? Is it clear to all of you? Right? Clear. Okay? Now, yeah, we will be discussing it again. Hold on. So, what now will happen is that, do you think it is very easy to, let's say, find the sum of the, is it a definition for this law? Nothing as such. Okay? There is no word to word thing that you have to mug it up. It is not our grade 9th and 10th. Okay? Don't worry. Just understand how to use it. That just should be sufficient. Okay? Now, what will happen is that, do you think it is very easy to add the two vectors by using triangle law or let's say, parallelogram law and get the exact value? If I already draw, let's say, triangle on your notebook, then it is very easy. Okay? But suppose I tell you that there is a velocity of 2 meter per second in one direction. Then in 30 degrees south of east, there is another velocity having 22.5 meter per second. So, find out the sum of these two velocities. Will you be able to draw it very accurately, geometrically or this method is very error prone. What do you think? Even after you draw it, you have to draw a straight line, measure the length of that straight line. Will it be error prone or not? It will come with a lot of errors. You have to be very, very careful. It will take a lot of time to draw the perfect triangle, perfect parallelogram, find out the angle perfectly, then getting the length of the sum of the two sides, measure with this scale very, very accurately. Do you all agree or not? Type in. So, what do we need to get rid of such errors? We need an equation. Geometrically, things will be better to understand. But when it comes to exactly finding it out, you need an equation. Just like you remember, we discuss about the calculus. We discuss something similar. Geometrically, the slope of velocity time graph is acceleration. But do you think finding the slope is so easy? Drawing the graph, finding the slope, angle, tan of that angle, it becomes so messed up. Similarly, here also, graphically, things look very rosy and straightforward. But practically, when you do it, it becomes a headache. So, we need an equation to exactly find out. All of you keep writing with me, find out the sum of two vectors in terms of their magnitudes and the angle between them. Anyone has any doubts? Type in. What's after we need an? We need an equation. This is a short end for me. This is an equation. We need an equation. Is it not good? You don't like it? Equation. I write perpendicular like this, parallel like this. These are short ends I use. These are the things that my dad used to use, these kind of things, when he was teaching me. All right. Anyone has any doubts till now? No doubts. What I want? I want an equation. Equation for finding what? I want to find out the equation so that I can find the length of the sum of the two vectors in terms of the length of both the vectors and the angle between the two vectors. Anyone has any doubts? Quickly type in. No doubts. So, before I get into, no, let me do that only first. Then we'll go to the questions. But there was a very good question that I could have done earlier, but we'll do later. Not a problem. I will not skip it. Don't worry. So, here is the situation. In a school exam, in a school exam, they will refer it as cosine law, derivation. It's an important derivation for your school exam. All right. Was it done in NPSHSR? This derivation that I am going to do? Is there any other school that is done with the vectors already? Okay. In DPS. Good. Good. So, you guys are already experts of the vectors. Must be very confident. Okay. So, let us say this is vector A and this is vector B. The angle between the two vectors, how do you represent? This angle is right. Right? We can say it is theta. You can say it is phi. Whatever comes in your mind, you can write anything. Okay. So, in order to add them up, should I create parallelogram? I have to create parallelogram. Right? When I create a parallelogram, it becomes this as the sum of the two vectors. Okay. This one. This is, let's say, vector C, which is A plus D. Okay. So, we need to find out, we need to find out the length of C in terms of, as a function of length of A, length of B and theta. Okay. Now, let me tell you this derivation has nothing to do with whatever you have learned today. It is a geometry. So, try to find out the length yourself. Okay. If you have done it in school, then also try to do it yourself here so that later, you know, you never have to practice it again. Okay. Yeah, Anika, I am not saying anything. I am waiting for you to find out. I am waiting for everyone to find out. Okay. Now, tell me, do you know any law? All of you listen to me. Do you know any law which connects the length of one side with the length of other two sides? Any law related to any triangle? Did you know already? Pythagoras theorem? Great. Great. Okay. So, that is our starting point, because here also we need to find length only. Right? We want to use Pythagoras theorem here. But what is the problem? Understood, Swara? We want to use Pythagoras theorem here. But what is the problem? We cannot use it. The reason is what? What is the reason, everyone? What is not there? 90-degree triangle is not there. There is nowhere it is 90-degree. So, if 90-degree is not there, what do you do? What do you do if it is not there? You create 90-degree. You create a 90-degree. So, let us create the 90-degree. What you can do? You can extend this and you can drop like that. Okay. This is the 90-degree. Now, how many right angle triangles you see here? Everyone, how many right angle triangles you see here? Two right angle triangle? Very, very good. Let us say this is 1, this is 2, this is 3, this is 4. What is the name of the triangles? Triangle 1, 3, 4 and triangle 2, 3, 4. These are the two right angle triangles. So, I can use these two right angle triangles for Pythagoras theorem. This is the 90-degree. So, although I can use it, but I cannot randomly take anything. So, let us say 3 to 4 length, this length is let us say L1 and this length is L2. Then, can I say that the sum of the two vectors, the length of that square is equal to L1 square plus L2 square. Can I say that or not? Type in everyone. I can say that and there is nothing wrong with it. The length of C, you already got it. Root over L1 square plus L2 square, there is nothing wrong with it. But can you leave it like this? No. Why? Because I want in terms of A vector, B vector magnitude and angle theta, this is in terms of L1 and L2. So, what do you do next? You find out L1 and L2 in terms of magnitude of A, B and angle theta. So, let us try to do that. So, this angle is what? Everyone, what do you think this angle is? Theta? This theta? Very good. This length is what? 2 to 4 length is what? 2 to 4. Mod of B, this is the length. So, what do you think L1 should be? What is L1? There is a right angle triangle. So, what is L1? In terms of B and theta, can you say something? Correct. B sin theta. All of you get this or not? Type in perpendicular upon hypotenuse sin theta. So, B, hypotenuse sin theta is perpendicular which is L1. Type in is this clear or not? Everyone? All of you, quick. Now, tell me what is this length? 2 to 3 length is what? B cos theta. So, do we all agree that length L2 is length of A plus length of B into cos theta? Isn't it? So, the length C is root over. Now, substitute the values B sin theta whole square plus A plus B cos theta whole square. This is what it is. Now, quickly simplify and let me know what you get. This has open the bracket, simplify as much as you can and let me know. Okay, this will be magnitude of B square sin square theta plus A square plus magnitude of B square cos square theta plus 2AB cos of theta. Okay? So, if I take B square common from this and that, it will become sin square theta plus cos square theta? Getting it or not? So, ultimately, you will get this length as root over A square plus B square plus 2AB cos theta. So, this is the cosine law. Why cosine law? Because there is a cos inside. Okay? So, this is what it is. Is it clear? Quickly type in will be asked the derivations in your school exam definitely 110 percent they may ask you. I think NPSR's UT happened right. How many derivations came? In NPSR, which one came? Raghav, equation of motion. Calculus method or graphical one? Graphical. Okay. Alrighty. So, this is the cosine law. Now, is finding the length of is finding the length of the vector sufficient or do I need to also talk about its magnitude? Sorry, the directions. What we got is just a magnitude. Right? Should we also talk about the magnitude or not? Should we also talk about the direction or not? Definitely, we need to discuss the direction also. Right? Direction in terms of what? In terms of what? Direction in terms of same AB and theta. So, let us quickly talk about the direction. So, what is the best way to find the direction of vector C? What you can do is that you can find out this angle. This angle, let us say phi. Alright? Is there a way to find out tan of phi? Can we find out? What is tan of phi? Quickly type in. Tan of phi is L1 by L2. All of us agree or not? Perpendicular divided by the base. So, this is B sin theta divided by A plus B cos theta. So, this equation will help us to find out the tan of the angle between the some of the two vectors and A. Okay? Suppose I have to find tan of angle between some of the two vectors and B. Then what do you think tan of this angle will be? Tan of alpha would be what? Just you do not need to derive it. Quickly tell me. If tan of phi is this, this is what angle between C and A. Suppose I have to ask you angle between C and B. Then how will you write? No, no, no. Not like that. Can you quickly modify this equation? This is angle between C and A. C and B what will be the tan of the angle? No one? Okay. Tan of the angle between C and A would be instead of B you have to write A. So, A sin theta divided by B plus A cos theta. Clear or not? Everyone, you can do the derivation also, but why to derive everything? It is analogous. Type in making sense or not? Everyone, type in, type in. Quick. All right. So, shall we take a break now? We will meet after the break at 7pm. Come back in time. All of you. All right. Are you able to hear me? Everyone, am I audible? So, let us continue. Let's continue. So, finally, we have an equation. If you tell me that there are two vectors, one of this much length, other is that much length, angle is this much, I will exactly be able to tell you that what is the length of some of the two vectors. Okay. And we don't need to draw the graph or anything like that. So, let us try to analyze this cosine law a little bit more. What I am trying to say is this. We have got that the sum of the two vectors is A square plus B square plus two AB cos of angle between the two vectors. So, when I just write A, I don't put a bar at the top, then this A by convention, let us assume that is the magnitude of vector A. And similarly, B and C also. Otherwise, this is little uncomfortable to write and C also. So, this is the sum of the two vectors of vector A and vector B, which has an angle theta. So, can you tell me if we can change the angle theta, then at what angle the length will be minimum? The sum of the two vectors length of that would be minimum at what angle? What do you think? Everyone, 90. What do you think? I mean, do we all understand that for this cos theta, whenever cos theta is minimum, that would be leading to the minimum value of C or not? Yes or no? Cos theta whenever it becomes minimum, that will give us minimum value of C. And what is the minimum value of cos theta? Minimum value of cos theta is not 0, minus 1, minus 1. And when that happens, when theta is 180 degrees, we have learned this in technometry. So, this is equal to root over A square plus B square minus 2AB, which is root over A minus B, the whole square and that gives us A minus B. A minus B. You have to make sure that square root term, whatever comes out is a positive quantity. So, it could be A minus B or B minus A depending on which one is greater. What do you think the maximum value will be? All of you, what would be the maximum value when theta is 0 and cos theta is 1? So, that will be root over A square plus B square plus 2AB. So, this is root over A plus B, the whole square which is A plus B. So, the maximum value of some of the two vectors can be A plus B. The minimum value of some of the two vectors can be A minus B. I hope it makes sense. Now, if you put theta as 90 degree, what does cosine law become? Cos of 90 degrees, what? Cos of 90 degrees, everyone 0, 0. So, you get C equals to root over A square plus B square or C square is equal to A square plus B square, which takes the shape of Pythagoras theorem. So, Pythagoras theorem comes from the cosine law only. Pythagoras theorem is a special case of the cosine law. Clear? All right. So, this is about the how you add the two vectors geometrically. So, all of you try solving this numerical. Everyone, I want to find out what is A plus B plus C plus D plus E plus F. What is this equals to? Just a quick numerical here. Try to find out. Anyone? No one? Should I do it? Should I do it? Sir, one second. Okay. Someone got something. Others? Okay. Nobody else? All right. Listen here, everyone. Let us see A plus B. Look at the geometry. A plus B is where? This is A, this is B. Where is A plus B? Do all agree that this is A plus B? Everyone? This is A plus B or not? Type in quick. Everyone, some of you looks like haven't come from the break. Right? This is A plus B. Now, if this is A plus B, do we all agree that this vector is A plus B plus C? This is A plus B. You add C. This will become A plus B plus C. All of us agree? A plus B plus C is this. Now, you add D. This will become A plus B plus C plus D. This vector is A B C D. Agree or not? And then, finally, add E to it. So, this vector will give us A plus B plus C plus D plus E. This one. All of us agree? Now, whatever comes out A plus B plus C plus D plus E, do we agree that this is equal and opposite of F? Equal and opposite of F or not? This is equal to negative of F. When you take this F that side, sum of it becomes zero. Sum of it becomes zero. So, this brings to us an important law. Suppose this is not a hexagon. Suppose this pentagon, then if I arrange vector like this, will the sum of all the vector be still zero or not? So, no matter how many sites a polygon has, no matter how many sites a polygon has, if the vectors are arranged in the same order, cyclic order clockwise or anti-clockwise, if you add them up, it will give you some as zero only. Yes, R A and C. Listen, this is A plus B. Head to tail. Now, this is A plus B plus C. This is A plus B plus C plus D similarly and this vector is you add E to it and this vector is negative of F. So, that is what we did. Clear? You can watch the video also once. This we already did. So, we will take a pause here. We will not talk about multiplication right now. I am going to tell you something more important, relevant to the physics and then I will talk about the multiplication. So, what we learned till now was some of the two vectors and we learned the cosine law. Now, it looks very comfortable that you can add the two vectors, you have an equation and you get the answer directly. Nothing wrong with it but suppose you have multiple vectors. For example, let's say you have this vector to be added to that one then added to this one and added to one more vector, let's say this one. This vector, this vector, this vector and let's say this one. I have to add four vectors. Can such thing happen? Will there be any situations where multiple vectors have to add, not just two? Do you think of? Can you think of any such? Tell me one such situation. You can unmute and tell. Okay, great. So, one object can be applied with multiple forces. Total force will be sum of all of them and forces of vector quantity. So, you have to add them like vectors. So, many a times you need to add multiple vectors like this. Many a times you have to do like this. So, what we know is addition of the two vectors. So, if you want to use cosine law over here then it will be a lot of problem. For example, first you have to add let's say vector a with vector b. You have to understand what is the angle between a and b before adding up. Then only you can use cosine law. Then after adding a and b, whatever comes out, let's say c, that vector whatever comes out from a plus b, what angle it makes with the third vector, then add the third vector by using cosine law. Then whatever comes out by adding the third vector, find the angle of all of that with the fourth one and then again use cosine law. Do you think it is easy or it is hell lot of calculations? What do you think? It has lot of calculations. I have to add two vectors, whatever comes out, angle of that with the third one. Then add the third one using cosine law, find the magnitude, the angle of everything with the fourth one. So, it becomes a very repetitive kind of thing. Now I am going to tell you another way of adding it up. So, you will appreciate the power of vectors for the first time. All of you listen very carefully, especially those who are learning vectors for the first time. I am trying to draw the right angled triangle like this. So, basically what I am doing, I am trying to create a triangle for all the vectors in two specific directions. One is let's say along the y-axis and other is along the x-axis. So, once this triangle is completed, now let me write it as a1 vector, a2 vector, all of you draw it with me, don't wait for me to complete first. I will erase as soon as I complete. This is d1, this is d2. Can I write a plus b plus c plus d? Can I write it as a1 plus a2 plus d1 b2 plus c plus d1 plus d2. Can I write it like this or not? I am saying a is a1 plus a2. So, instead of a, I can write a1 plus a2. Yes or no? Type n. Everyone, what happened to everyone after the break? Everyone sleeps. Type n quick. I think break should not be given right after the break something happens. So, this is written as a1 plus b1. I am just rearranging things plus d1 plus a2 b2 plus c plus d2. I can do this. Now tell me adding vectors like this is easy or adding like that is easy. Which one is easier? Left hand side or right hand side? Okay. All right. Left hand side is a plus b plus c plus d directly. Many of you are saying, Ajay, tell me, don't you think adding these three vectors is easy? a1, b1 and d1. Everything in the straight line. Is it easy to add them or not? Why it is easy to add them? Because vectors are added like scalars when they are in one straight line. Similarly, adding these two is easy. Yes or no? Everything is vertical line. So, adding them is easy. Are you getting it? So, let's say velocity two meter per second here. You can add three meter per second velocity in the same direction. Total will be two plus three. So, they add like numbers. You don't need to use cosine law for them because everything is parallel or anti-parallel. If it is in opposite direction, subtract the magnitude. Same direction at the magnitude. Right? So, this will give us a vector which is in the horizontal direction. Yes or no? This will give us a vector in the horizontal direction or not? All of you agree or not? If I keep on adding vectors in this direction, I will get a vector in that direction only. Right? And this will give us a vector in the vertical direction. All of us agree. That will give us a vertical in vertical direction vector. This will give us a horizontal direction vector. Right? So, all in all, the sum of all the four vectors is nothing but sum of the two vectors which are perpendicular to each other. Now, how you add the two vectors perpendicular to each other? One vector is this, other one is like that. This you can say is V. This you can say is H. Now, if the best thing is if you create a parallelogram, if you create a parallelogram, the sum of the two vectors which is this resultant, the length is root over this length square plus that length square because this is also V. Do you all agree? Type in quick. Magnitude of the resultant is root over H square plus V square. Now, tell me cosine law is easy or this one is easy when you add the four vectors. If you have any doubts, you can type in, you can speak also quickly do that. This is lot easier. This is lot easier. Now, if this is lot easier, let us try to derive the cosine law by using what we have just done. And by the way, whatever we have just done is called taking components. Have you ever heard of taking components before? Components, what have you ever heard? Have you heard about it? Components? Yes, coming to that. Hold on. Hold on. Coming to that. Well, what I did before adding A with B, I have broken A into A1 and A2. So, A1 and A2 are components of A. Before adding D, I have broken B into B1 plus B2. So, B1 and B2 are components of B. All right. Similarly, C is already in one specific direction. You do not need to take component. And D, when you add D1 and D2, D1 plus D2, D1 and D2 are the components of D. Getting it all of you, you have to type in quickly. Otherwise, I get to know that I will even slow it down. I think none of you are understanding them if not everybody is replying. Understand or not? Okay. Now, it means that, what it means? It means that when you're adding the two or three vectors, breaking the vector first into its component and adding them, that is easier. But have you ever done it with numbers? Let's say 3 plus 5, have you ever written 3 as its component like 2 plus 1 and 5 as 4 plus 1? And then add, have you ever done such things? This is meaningless with numbers. It doesn't help us in any which way. But it helps a lot with vectors. When you break vector A into its component, D into its component, D into its component, then it helps us a lot. Okay. Now, we will take an example of just two vectors and I will use all of this to tell you how exactly we can find the sum of the two vectors. All right. So someone was asking how to find out the value of A1 and A2? That also I'll answer while discussing that itself. Fine. So let's say this is a derivation of cosine law by using components. This is A. This is B. All of you just keep noting it down. I'm not asking you to do anything. Just note down. All right. This is angle theta. Now, I have to add these two vectors. I don't know anything about the cosine law. Let us see. Then what I'll do is that I will, the first step would be, I will draw two perpendicular lines x and y. That is your step number one. Step number one is finding out which direction is your x-axis and which direction is your y-axis. You can take any direction as x perpendicular to that. Any direction can be y. All right. And I prefer to take everything in the plane of the diagram. So I have taken x this way, y that way. Fine. Now what I will do, what I'll do is that I will break A into its components because B is already along the x-axis. I don't need to break the B. I will break A. So A will be this vector plus that vector. Fine. This plus that. So adding A plus B is as good as, it is as good as adding vector B. This is B plus adding the component of A this way and component of A that way. Do you all agree adding two vectors A and B is as good as adding three vectors B, A1 and A2? All of you type in. Make sense or not? Right. Now tell me, now tell me what I'll add first. I'll first add B and A1 and then I'll add A2. So what is A1? Length of A1 is what? Okay, someone is saying A1 a part of B. Why are you saying A1 is a part of B? A1 is a different vector. A1 is component of A. I have just drawn on top of B but doesn't make it a part of B. B is a different vector. So A1 is what? What is A1? If this is 90 degree, the length of this vector, let's say A. So what is this length? Length of A1 is what? Everyone, won't it be A cos theta? Type in and length of that would be A sin theta or not? Everyone, magnitude of A2 is A sin theta, magnitude of A1 is A cos theta. All right. So I will first add up these two vectors. I'll add these two. So when I add these two, I'll get one big vector this way whose magnitude will be B plus A cos theta. All right. And I, this vector, this one would be A sin theta. All right. Some of these two vectors, you can find out by getting the length of this. Length of that is root over this square plus that square which will lead you to the cosine law only. All of you type in, make sense or not? This is your R. R would be by using Pythagoras theorem root over A sin theta, the whole square plus B plus A cos theta whole square which will give us cosine law only. Okay. All right. So when we add the two vectors, three vectors, four vectors, taking components always help us. Now you will appreciate more about the components when you solve numericals, right? That we will be doing as we progress through the physics chapters one after the other. You will see that we will be taking components many, many times. All right. Now I'm going to tell you some things which are very, very essential. So focus everyone. If you have a vector like this, okay, this is a vector, here is a vector, vector A. I want to find out its component, its component in the x direction and in the y direction. All right. Horizontal and vertical direction. So can you tell me what is the length of the component of A which is already vertical in the horizontal direction? Try finding it out. Okay. Let us say it is having a component in the x direction this much. All right. Now if this is the x direction component and you try to complete a triangle by creating a y component, will the triangle be ever be completed? If this is your x component and you draw the y component, will triangle be completed? The only way theoretically you can complete a triangle is doing what? Let's say this is A1 and this is A2. Only way the triangle can be thought to be completed if A1 is equal to null vector. The length of A1 should be zero. Okay. Clear. I hope this is clear. Now comes the biggest statement which will be very, very powerful when you do the physics numerical. So everyone put a star mark like this and write down a vector will have no components or no component perpendicular to itself. What does this mean? This means that a vector cannot affect anything which is perpendicular to itself. Now I will give you certain examples so that it will be even more clearer to you. So everyone write down force in x-axis as per the statement above. This force, can it create acceleration in the y direction? Can never create acceleration in y direction? Why? Because this force has no component in the y direction. This force cannot affect anything in the y direction. Similarly, the acceleration in the y direction can never change velocity in the x direction. That is the property of a vector. A vector cannot affect anything perpendicular to itself because it has no component in that direction. So these are just example number one, example number two. There can be millions of such examples. Now all of you, can you type in whether the sum of the finding the sum of the two vectors, the concept of that is absolutely clear. Yes or no? Sagar is saying no. Then he is saying yes. All of you type in till now everything is clear. This is all about, this was all about the how we add the vectors. Now we will learn about till when is the class by the way, till 8.15. So probably we will be done with the multiplication of the vectors and not entire vector can be done today. Probably next class we need to a little bit extend it. So now we will talk about how multiplication with the vectors is done. So everyone write down multiplication of vectors. So tell me can I multiply scalar with a vector? Can it be done? Can I multiply scalar quantity with a vector? Those who are saying yes, give me some example where scalar quantity is multiplied with a vector. No, that is not correct. Equation of motion when you force equal to MA, when you write everyone, when you write force is equal to mass times acceleration, everyone. Mass is a scalar quantity or not? An expression is a vector quantity. So is it a multiplication? Yes or no? Is it clear Vishnu? There can be so many examples. Like somebody said v is equal to u plus a t. You are multiplying acceleration with time or not? Time is a scalar quantity. This is a vector equation by the way. Can you tell me is it possible to add scalar with a vector? Is it possible to add a scalar to the vector or that doesn't mean anything? Everyone, it does not mean anything. You can never ever add a scalar with a vector. It doesn't mean anything. So if we multiply scalar with a vector, will we get a scalar quantity or a vector quantity? What do you think? We will get a vector quantity only. So write down outcome is a vector. Let's say the scalar is lambda. If a scalar is lambda and the vector is a. So the scalar times vector lambda a, this thing, will the direction remain same? At least direction remains same. Type in yes or no? When you multiply scalar with a vector, the direction will remain same or not? My dear friends, everyone, I mean, just answer with right or wrong. Doesn't matter. Participate. Participation is important. It remains same or not? Okay. Now what if lambda is less than 0? If it is lambda is a negative quantity, will the direction change or remain same? Let's say lambda is minus 1. Minus 1 is the scalar quantity. When you multiply minus 1 with a, vector direction changes or not? Changes or not? If a become minus a, are you telling me some of you a and minus a, the direction is same? It will become an opposite direction. When you multiply negative scalar, the vector switches to the opposite direction. Okay. Somebody is saying, how can a scalar quantity be negative? Do you do anyone know any quantity that is negative scalar quantity? Quickly, any quantity that is negative, which is scalar? Everyone charge. Charge can be negative. Displacement is a vector quantity. Charge is a scalar quantity. Charge be negative or not? Charge can be negative. Energy can also be negative. Length cannot be negative. Does that answer the question? Whether the scalar quantity be negative? Right? So, it can be negative. There are so many examples. So, lambda less than 0. What happened to the direction? Direction reverses 180 degree. And my dear friends, if lambda is equal to 0, what happened to the direction? What happened to the direction? Direction vanishes. It becomes a null vector. Only if lambda is greater than 0, the direction is same. Okay. This is about the direction. Now, what about the magnitude? When we find the magnitude of lambda times a, will magnitude be greater than a or not? Magnitude of lambda times a. Lambda times a is a vector. So, direction can reverse here and there, whatever can happen. But what will happen to the magnitude I am asking? Will magnitude change? And if it changes, will it increase, decrease, remain same? What will happen to the magnitude? All of you, magnitude will increase. How many of you say that? Okay. Saket is saying lambda is greater than 1, then it will increase. If lambda is equal to minus 2, will magnitude increase or not? If lambda is minus 2. Everyone, I am asking you, if lambda is minus 2, the vector become minus 2a. So, minus 2a, the magnitude of minus 2a is more than the magnitude of a or less than? Okay. So, now, I ask you every time some trick question. So, lambda times a is basically magnitude of lambda. Can magnitude be negative? It can never be negative, magnitude of a vector. So, you have to take mod of the lambda also. If lambda is greater than 1 or lambda is less than minus 1, the magnitude will increase always. But if lambda is 0, magnitude is 0, that sounds straight forward. Now, if lambda is between minus 1 and 1 and not equal to 0, what will happen to the magnitude? If let us say lambda is 0.5, magnitude increases, decreases or becomes 0, decreases, magnitude goes down. So, this is what happens. Does that answer all of your questions? Anyone has any concern here? Note it down quickly. Everyone noted down. All right. So, shall we move? Shall we move? Last one, take an example. Let's say lambda is equal to 0.2. Don't you think when you multiply 0.2 with magnitude of a, magnitude will go down? Yes or no? So, that's what? Between minus 1 and plus 1, if you multiply a number with a, it will decrease the magnitude. Right? Now, we are done with multiplication of a scalar with a vector. All right? Now, can a vector be multiplied with another vector? What do you think? Can a vector quantity be multiplied with another vector quantity? Give me some example. Some examples. Okay, good. Somebody has written what we got v square equal to u square plus 2 a s. Very good. Actually, the a and s are these vectors. So, there is a multiplication of acceleration and displacement. Good. Then you have work done. Do you remember this work done is a multiplication of force and displacement? Two vectors? Yes or no? Right? Do you remember any other equation? Do you remember any other equation? Tell me. No other equation? All right. So, there is going to be many equations that you will see where one vector is multiplied with another vector. When you say momentum is m into v, m is a scalar quantity, my dear friend. We are looking for vector multiplied with another vector. Right? So, there are equations like torque is equal to r multiplied with another vector, this, like that. There are many such equations actually, but at least one such equation you should know this one and one such this one. And look at this. Two vectors are multiplied. What comes out? What is r? Forget about that. R is just a vector. All right? We will learn about it in detail, but there is such equation. All right? You can see two vectors are multiplied. What are you getting? Are you getting a scalar quantity or a vector quantity? What is the work done? Work done is a scalar quantity or a vector quantity? Everyone, it's a scalar quantity. Scalar quantity. Right? Two vectors are multiplied and you are getting a scalar quantity. Here you can see that two vectors are multiplied. What are you getting? A scalar quantity or a vector quantity? Vector quantity. Right? So, scalars will be multiplied only in one way, but vector multiplies in three ways. One, the way it multiplies with another scalar quantity, a vector multiplied with a scalar quantity. The two ways in which a vector can be multiplied with another vector, one will lead to a scalar quantity, other will lead to a vector quantity. Okay? So, this is what it is. Please note it down. Dot product is a name of the multiplication of a vector with another vector that leads to a scalar quantity. Cross product is a name that leads to a vector quantity. Alright? All of you written, I'll go to the next dot product. Let us say I have to multiply a vector with another vector and outcome is a scalar quantity. Then dot product, the name is also called as scalar product. There are two names to it. It is written like this, a vector put a big dot between a and b. This is how it is written dot product. Alright? The outcome is magnitude of a into magnitude of b into cos of angle between a and b. Clear to all of you? Now, you may ask from where the cos theta came? Why not tan theta? Why not seek theta? Why not cos square theta? For that, I don't have any answer because this is the way people have assumed it to be. A dot b is defined. It is defined. Defined means it is assumed. It is assumed to be magnitude of a into magnitude of b into cos theta. Alright? Now, why this is a logical choice? I can explain that to you. But this is the formula for the dot product. Alright? Do anyone has any concern here in this definition? Type in any issues? No? Alright. So, let me tell you why it is good to write a dot b as a into b cos theta. Alright? So, let us try to understand what is this? What is this b cos theta? Why at all technometrics function? Very good question. Let us see that. Alright? Everyone? So, this is a vector. This is b vector and this is theta. Now, b cos theta, where do you see b cos theta? If you drop a perpendicular and say that b is the sum of these two vectors, the length of this vector is what? The below one is what? This is b1 vector. This is b2 vector. So, magnitude of b1 is what? This is 90 degree. Magnitude of b1 is b cos theta. All of us agree or not? Type in. Everyone? Right? Now, this b cos theta, b cos theta, can I say it is the component component of b in the direction of a. I can say that. So, what I am doing? I am multiplying a with component of b in the direction of a. So, it means something. Similarly, the same diagram if I take here this, this is b and this is a, this is theta. Can you geometrically find out where is a cos theta? Because this thing can be written as b into a cos theta as well. Can you find out where is a cos theta in the diagram? Just like we did from b cos theta. Everyone, let me know once you are done. So, let us see. If I extend b like this and drop a perpendicular on b, this is 90 degree. Can you tell me how much is this length from here to here? This length is what? This length is a cos theta or not? Magnitude of a into cos theta? Type in all of you. a can be written like this and that. What is a cos theta? a cos theta is nothing but a can be written as this vector plus that vector. Right? So, a cos theta is a component of a in the direction of b. Okay? So, I will write a cos theta component of a in the direction of b. So, you can think of dot product as multiplication of the magnitude of one vector with component of another vector in the direction of the first vector. So, you can say it is a into b cos theta or b into a cos theta. Fine? So, I hope you understand why it is logically that. Fine? Now, do you remember that the work done is multiplication of force with displacement in the direction of force? Do you all remember that or not? Multiplication of force into displacement in the direction of force. That is the work done. It is not just force into displacement. Okay? That might not have been covered to you because you dealt with only the straightforward cases in grade 9. But that is fine. In grade 11, we will learn that in detail. All right? So, the formula for the work done, my dear friends, is force dot product with displacement. It is f into s into cos of theta. All right? Okay? So, this is called the dot product. I hope this makes sense to everyone. Type n. Is this clear? Everyone? All right. Let me show you. Let me show you the cross product now. You don't need to worry about this. We have already written enough about the dot product. Okay? This also we have written. All right? Same thing is in the slides. You only told me that don't show the slides. So, I am writing everything down. Now comes the cross product. Now, any guesses if a dot b is a into b into cos theta, a cross b will be what? Just take a wild guess. Very good. Very good. Now, haven't we learned that when we use the cross product, we should get a vector as the outcome? Yes or no? We should get a vector as an outcome. Right? Right? So, when you say a into b into sin theta, is that a vector or a scalar? Is that a vector or a scalar? You are telling a scalar thing. Right? But then that is correct only. What you are telling is the magnitude of the cross product. Okay? So, cross product or the vector product, let's say a cross b. This is how you write the cross product. Okay? Magnitude of a cross b is equal to magnitude of a into magnitude of b into sin of angle between a and b. Clear? Now, how can you think it as if a into multiplied by what? Component of b in which direction? Component of b in which direction? In the direction perpendicular to a. In the direction perpendicular to a. That is what a cross b magnitude is. In the direction of a, it is b cos theta perpendicular to a is b sin theta. Alright? So, this is a into component of b perpendicular to a. Okay? Or you can say b into component of a perpendicular to b. One and the same thing. Alright? But then this is about the magnitude of the cross product. What about the direction? It is not a scalar quantity. It is a vector quantity. We found out the magnitude of the vector only. Let us talk about the direction of the vector. Clear? Now, everyone, this is a, this is b. Alright? Direction of the cross product is given to b. Write down perpendicular to both a and b or perpendicular to the plane containing both a and b. Okay? It should be perpendicular to a and b. So, a and b together will be in a plane. So, right now in which plane a and b are in? Everyone? In which plane a and b are in? It is a plane of the screen. So, perpendicular to the plane of the screen. Which direction do you think it is? How many directions are perpendicular to the plane of the screen? One or two? There are two directions. One coming out of the screen. Other coming into the screen. Two directions are there. All of us agree, right? So, there is an ambiguity. Even though we define it to be, even though we have defined it to be perpendicular to the plane, but then also there are two directions to pick. You have to uniquely pick one direction and whatever you pick should be picked by other also. It should not be that you create a rule. By that rule, you get one direction. Other person gets the other direction. So, it will create an ambiguity. To remove that ambiguity, we will talk about a rule which we can use to uniquely assign a direction. Either inwards into the screen or coming out of the screen. So, this is the most important thing for today's class that I'm talking about. Listen to it very, very carefully. This will ease up your life like anything. Okay. You will be able to proceed with cross product very, very easily because the only thing where students make most of the mistake is finding the directions. All right. So, I want to find direction of A cross B, let us say. So, everyone take out your right hand. Even though you are lefty, take your right hand. Okay. All of you are done. Take out your right hand, everyone. Align in the direction of the first vector. In the direction of the first vector you align. In the direction of A, it will be like this. Okay. Like this you align. All of you align. Now, fold towards the second vector. You cannot fold like that. It should be natural fold. So, like this. Align in the first vector direction, go towards the second one. Thumb is telling you inward or outward. Thumb is telling you inward or outward. Outward. Outward. So, A cross B is outwards. A cross B. Now, it will be B cross A. Which direction B cross A will be? Same thing. Align in the direction of the first vector. Now, first vector is B. It is not A. B cross A. Which direction it is? Everyone. Align in the direction of B, fold towards A. Inwards or outwards. Now, it will be inwards. Everyone agrees to it or not? You aligned it towards B, fold towards A. It will be inwards. So, such a simple rule it is. And I have seen students complicated. They use like this. Okay. This is first vector. This is second vector. This is third vector. So, like this. They keep on doing like this. Okay. So, don't do like that. Just align the first direction. Go towards the other one. So, it will be B cross A. Now, do you all agree that A cross B and B cross A are in opposite directions, right? And they are equal in magnitude. So, can I say that A cross B is equal to minus of B cross A? Can I say that or not? Equal in opposite vectors. Type in. Can I explain outer inward? Listen to it. I will take one example and explain. But right now you write it down. Whatever is written. One last example I will take before leaving you. How many are attending the KVPY classes? Okay, some of you are attending. I hope it is going on very well. And yeah, it is hectic. But then what to do? You have to complete the curriculum, right? You have to complete the curriculum. So, we will not tell you something which is not true. We will not tell you that things are very easy, even though they are not. So, whatever they are, however they are, you have to do it. If we want to aim at something, right? Let us take one final example. Everybody, everyone, one final example, then I will leave you for the day. Somebody else will start teaching you after 15 minutes. Hold on. So, let us say this is a situation. This is A vector. This is B vector. Tell me the direction of A cross B. Where it will be? Inward or outward? A cross B. Others? Inwards. It will be inwards. Align your hand in the direction of A. Go towards B. Align direction of A. Go towards B. All right. Okay. Fine. Many a times you will see vectors like this. They are arranged like that. But first, you have to connect them tail to tail before finding out the direction of cross product. Move this vector parallel to itself. Connect it tail to tail and then use this law. Don't use this law for head to tail connection. So, this is A. This is B. Okay. So, put it like this and then align the direction of A. Go towards B. A cross B now is outwards. All right. All right, friends. So, that's it from my side. We will now end this session and we will continue with the vectors in the next class and complete it. Little bit is remaining. All right. That's it from my side. Bye for now. Thank you, sir. Thank you, sir. Bye. Thank you, sir. Bye.