 Hello. Are you guys there? Anand and Arreman? Yes, sir. I'm here. Yeah. Okay. Fine. So there are a couple of minutes. We'll wait for everybody to join. Did you get the syllabus? What is coming in your next exam? Physics? Upcoming? Sir, from who? From you or from our teachers? From our school teachers. I don't know. Sir, my teacher hasn't told us anything. So we have one unit test or like formative assessment, whatever you want to call it. It's a lab. That's on 10th October though. They haven't told us what our assessment portions are. Okay. What all your teacher has covered in your school till now? In physics? We just did vectors and scalars and now we're starting kinematics. Okay. Anand, do you hear the music outside? What? Yeah. Dude, it's so loud. The bass is actually shaking my pillow right now, no joke. Oh, did they and some should join? Did you guys do the homework? Yes sir, I do submit to you. Since it is a line class, you can always take a pic and send it over to me as in WhatsApp images sort of. Oh sir, there's a private file function that I can send you, so I'll just put it onto a Word document and give it to you. What did you say? Sir, in the private messaging in the Zoom, there is like a slide file function that I can... How are you doing your homework? By the way, I did not receive actually. How are you supposed to send it? I mean, don't be very formal about it. You can just take a snap of whatever you have done. What's up? Okay, it's like an sending image. You know, there are a couple of scanning apps that are pretty cool, like cam scanner. Okay, so you can take a snap of all whatever you have done, a single PDF get generated out of it and then send it across to me. Both of you have done? Yeah, I'm done. Okay, I'm assuming you guys don't have any doubts? No. Okay, straightforward assignment. I did not send the other assignment that was on errors and uncertainty because I was in middle of it and in the assignment there are questions from all the topics of errors. Sir, since I did the homework digitally, I didn't really elaborate that much on the answers and some of the formatting might be a little off depending on which application you use. You're playing background music while talking? Sir, there's some party going on outside, sir. It's not a party, it's for Ganesh. Dude, the kind of music they're playing feels like a party only. You're asking me how you will send me? Yeah. Yeah, same. Just text, you can, what's at me? Whatever you have done or skid a PDF out of it using CAM scanner app. Okay, all the images, you take a... I sent it on the group chat. I'm not group chat, but like private chat. Yeah, I can see that. Sir, the formatting might be a little off. Understood. See, this is your school level homework. So in school, you get step marks also, right? So you cannot actually skip these steps, because at times you know exactly how to solve, but if you skip the steps, you'll not get the full marks. Okay, sir. Sir, I just had a mind to you. Can you check to make sure you got it? Yeah, I have received yours. So not sure whether... Okay, people might be celebrating Ganesh Chaturthi. Okay, so let's proceed. Okay, we cannot wait more than couple of minutes for one and a half hour class. Fine, so let me first share my screen. Are you able to see my screen? Yeah. Fine, so last class, we have done introduction of it. Okay, sir, I'm just gonna... Somebody came, Aditya came. Should I continue? Sure, sir. Okay, I heard something. Anyways, so last class, we have done a basic introduction of errors. All right? And when we define error, we are quantifying it with uncertainty. All right? As in, I don't know how much is the error, but what I know is what is the maximum amount of uncertainty on whatever I have measured. So whatever reading I take, I also represent how much maximum uncertainty I feel it has. So if I measure 2 with an uncertainty of 0.1, I'll write the reading as 2 plus minus 0.1. Okay? Now, these are the cases when you are directly measuring whatever you have to deal with. For example, you have to deal with length. Okay? So you are directly calculating the length and writing your readings with uncertainty. And then suppose you have to measure time. Time also, you can have a stopwatch and you can measure it with some uncertainty. Okay? Now, suppose you have to calculate velocity based on whatever length you have measured and the time you have measured. Okay? Suppose you have to deal with the average velocity. So you take a ratio of velocity and... Sorry, you take a ratio of length and time. Yes or no? So velocity, when you write, you'll write it as length which you have measured and call it as distance divided by time. This will give you average velocity. Okay? The average velocity is this. Now, this, you can calculate for the actual readings, but what happens to the uncertainty? I know the uncertainty in distance. I know uncertainty in time, but what is the level of uncertainty velocity has? Okay? So we must know as in how these uncertainty will combine. Okay? Somebody left. R M on left, is it? He'll probably join again. Okay, so velocity is distance by time. What I was saying was that I know what is the uncertainty in distance and what is the uncertainty in time. I want to know uncertainty in velocity. Okay? So I should know that if I take ratio of two readings, how the uncertainty, what happens to the uncertainty? Okay? Similarly, if my total length is the sum of two lengths, whatever I have measured, I'm not measuring the total length directly. I'm measuring L1 and I'm measuring L2. Okay? I know the uncertainty in L1 and uncertainty in L2. What is the level of uncertainty capital L has when I add the two readings? All right? So I must know that how the uncertainty will interact when you add the two. No, can you do the L thing again? I'm sorry. I'm a bit confused. Okay. L1 plus L2. Yes. So suppose L1 is let's say two centimeter plus minus point one and L2 is four centimeter plus minus point two. You have measured the two lengths. Okay? And your total length is sum of the two lengths. So you'll say six and then plus minus what? Right? Yeah. I should know how these errors will interact when you add. And suppose I want to find the difference in length. Suppose I need to find L2 minus L1. Okay? So readings are very easy to subtract four minus two is two. But will the errors will get subtracted or what will happen? Right? So that is what I need to understand here that what how the errors or uncertainty will interact. Okay? So this is what we are going to do next because I cannot measure everything directly. I'm calculating it. Suppose I have to calculate velocity. I may measure distance with some uncertainty, time with some uncertainty, but what about So can you show us how to do it? That's what I'm talking about. Yeah. Okay. Let me finish. Yeah. So what I was saying was that that is what this topic is about as in how can I deal with the uncertainty when I'm not directly calculating it. But I'm using the cal, I'm using the readings which I have taken. Okay? So I should know what happens to the error when you add or subtract. So please write down addition and subtraction. Okay? So let's see how the uncertainty will interact. If I add or subtract first. Now, suppose I have a reading for a which is which is X plus minus delta X. Okay? And the reading B is let's say Y plus minus delta Y. Okay? Now, I want to find out the sum of the two readings. I want to do a plus B. Okay? So when you do a plus B, of course, it will happen X plus Y. But what will happen to these uncertainty? Can anyone guess how much uncertainty Y will have? Delta X plus delta Y. Should I write like this plus minus delta X plus delta Y? Is this correct? I think so. Reason. There should be reason for it. Anyone want to talk about the reason for this? So, but then for it should be the same for minus also, right? I'm talking about plus here right now. Yeah. To the minus. Tell me why it should be plus delta X plus delta Y. What is the reason for it? Because see the error could be on both sides point one. I don't know exactly whether it is point one. Suppose delta X is point one. Okay. I don't know whether it is my actual reading is less than point one or more than point one or it is just less than I mean this is the maximum amount of error right point one. So it could be just more than point zero five or less than point zero five. Okay. So basically every time I write maximum possible error. Fine. That is why I'm adding the two maximum possible errors to arrive at the maximum possible error of Y. Okay. So addition is straightforward. Now suppose I have to do a minus B. Now it will clearly come. Reading will be subtracted X minus Y. Now what Aditya was saying that your uncertainty will get added up. It will never get subtracted. The reason for it anyone. So when you have two values of uncertainty uncertainty will not decrease. It will just increase. But then I'm subtracting not two values. So why not uncertainty will get subtracted. Like those are not the values you are let's say kind of looking for. So I mean I do not explain it. I'm sorry. Okay. Arama you were saying something. But you want to comment anything. Araman or Araman or Anand. Why it should be delta X plus delta Y. When you subtract the two readings. So because you can't remove uncertainty. You can add them together. That is fine. But then you are actually your operation is subtraction. Right. Yes. Like basically it's like saying that when you subtract the uncertainties is basically removing a possibility. And the thing is that whenever you and since uncertainties are something that happens to the original number subtracting or adding doesn't matter. Okay. So let me put it like this. Suppose uncertainty is point one delta X is point one and delta Y is suppose point two. Okay. I don't know exactly whether it is X is less than point one or more than point one. Similarly, I don't know whether Y is more than point two or less than point two. If I know exactly then I'll subtract and tell the exact value. So it may happen that X is less than or less by point one and Y is more by point two. So when you subtract the variation could be point three. Are you getting what I'm trying to say here? It could be that X is X minus delta X and Y is Y plus delta Y. There is a possibility that this happens. Okay. So maximum possible error when you subtract will become delta X plus delta Y in this case. Have you understood now? Are you getting it? Okay. So this is with respect to subtraction and addition. Your absolute errors will get added up. Please write down. Now let's see what happens when you multiply or divide. Wait. So this thing when you say X minus delta X Y plus delta Y, what are you trying to like? What was this for? Hello. See what I was saying was division. See, I'm saying that I don't know whether X is less than, less by delta X or more by delta X because it is error. It could be both sides. So it may happen that X is actually X minus delta X and Y is Y plus delta Y. The error can be of both sides. Yeah. When you subtract, then the maximum possible error will happen. Then will be delta X plus delta Y. When you subtract these two, it will be delta X plus delta Y. Correct. Okay. Now let's see what will happen if you multiply. Okay. So similarly we will take X plus delta X and Y plus delta Y and we will multiply these two ratings. So all three of you X plus minus delta X and B is Y, small Y plus minus delta Y. Now I want to multiply A and B. Okay. So can you write like this and open the brackets and tell me what you are getting? Is this what you are getting? Which I am writing here? Yeah. Okay. Now usually the errors are very small compared to the actual reading. Suppose you are calculating a length of 2 centimeter. You will not like to have an error of 1 centimeter. So it's like 50 percent error. Okay. So that is like unacceptable in the modern era. Okay. So we will assume that delta X and delta Y are very small. Okay. Delta X and delta Y, they are very, very small compared to X or Y. Okay. So that is why the multiplication of two errors I can ignore. It is close to zero almost. Okay. Because delta X is very small. Delta Y is also very small. So when you multiply two very small quantities, it will be near to zero. Okay. So I can approximately write this thing as Y is equal to X Y plus minus X delta Y plus minus Y delta X. Now let me divide everything by X Y. So Y divided by X Y is equal to 1 plus minus delta Y by Y plus minus delta X by X. Okay. Now take Y this side. You will get Y minus X Y divided by X Y is equal to plus minus delta X by X plus minus delta Y by Y. Are you getting what I am doing here? Yeah. Okay. See in your book it is directly written the final expression. I am helping you to visualize actually from where it is coming. There is always a good idea to know. Sir, I didn't really understand how the delta X and delta Y just crossed out. How delta X and delta Y crossed out. First step. Okay. This one. See I am saying that delta X is very small. Delta Y is also very small. Just like delta X let's say 0.1. Delta Y is also 0.1. Okay. 10 and X could be 0.1. So 10 into 0.1 is 1 but 0.1 into 0.1 is 0.01. So compared to 1, 0.01 is 100 times less. So I am ignoring this value. Okay. It is an approximation I am making here because usually the errors are very, very small quantities. Okay. Now Y minus X Y is actually delta Y. Y is the actual value X into Y is the multiplication of your two readings X and Y. So Y minus X Y is delta Y. So delta Y divided by you can say that X into Y as Y roughly. Okay. So this will come out to be delta X by X plus minus delta Y by Y. All right. So when you multiply the two readings, you can say that the relative errors are getting added up. Yes or no? Please write down. Relative errors gets added when two readings are multiplied. Any doubts? Quickly tell me. Any doubt? Aditya Anand? No. No. Okay. Fine. So division is I will not get into the derivation of how it happens in division. Okay. In your textbook also it is not there. So let me not put extremely mathematical derivation in front of you. So when you divide the two readings, for example, if you have two readings, A is equal to X plus minus delta X and B is equal to Y plus minus delta Y, then if you have to divide the two readings, okay, then this division usually happens, for example, you have to find average velocity. So A could be your distance and B could be your time. So when you divide the two readings, what happens to error is exactly same as when you multiply the two readings. Delta Y by Y is equal to delta X by X plus delta Y by Y. Fine. So this you must remember when it comes to the error, the subtraction and addition are same as in you have to deal with in a same manner and when it comes to division and multiplication, they are also same with respect to each other, how you deal with the errors. So here also relative errors in the two readings get added up, okay. Now I'll ask you a small question here. Tell me, suppose you have A is equal to X plus minus delta X, okay. Now you want to calculate A square. So how much relative error delta Y will have? So delta Y by Y will be how much? How much will be delta Y by Y? So I'm not sure. Anand. Is it 2 delta X? Only 2 delta X. 2 delta X by X, sorry. See, A square is nothing but you can take it like A into A. Okay, so when it was A into B, it was delta X by X plus delta Y by Y. Now, if it is A into A, you need to add the error in A two times delta X by X plus delta X by X. That will come out to be 2 delta X by X. Okay. Understood, Adya? Yes, sir. Okay. So like this only you have to deal with exponents. So suppose you have Y is equal to A to the power 6. Then same way delta Y by Y will be equal to 6 times delta X by X. Are you getting it? Whatever is in the power comes in the front. And that happens not only with integers. Even if it is a fraction, suppose Y is equal to A to the power 3 by 2. Then what will be delta Y by Y? Aditya? Yes, sir. So it's 3 by 2. So yeah, 3 by 2 delta X by X. Okay. Is that correct, sir? Yes. Remember this. Whatever is in the exponent comes in the front. All right? Arima, you're coming and going. So electricity keeps coming and going. So the engine also invariably comes and goes. Are you able to understand what we have done? Yes, sir. So you're trying to derive and explain how to calculate the uncertainties in addition to traction multiplication. Okay. So you have whatever is written in front of you, you have understood this, right? I don't need to repeat. Yes, sir. Okay. I think we've done this before in a previous class. I don't think so. Have you done it? Sir, for me and Anand, we did it in the first or second class. No, we didn't do this. We haven't done it. I think in your school it was done. What we have done last class was an introduction of error. What is absolute error? What is relative error percentage error? That's it. Okay. Now I'm going to write here few problems. Suppose y is equal to a square b to the power 3 by 2 divided by c to the power 5. Okay. Error in a is delta a, error in b is delta b and error in c is delta c. You need to tell me what happens to the error. Delta y by y will be what? Take some time and then tell me. Take a minute. Sir, I'm just guessing. It is okay. I mean, you're not expected to get everything right. So it will be 2 delta a by a plus 3 by 2 delta b by b. I don't know if it will be plus or minus 5 delta c by c. Correct. So remember a thumb rule. You never subtract errors. Okay. Even if it is like this, you know that x plus minus delta x is equal to y plus minus delta y. Then if you bring y this side, it will become x minus y and it will be plus minus delta x plus delta y. Even if you change sites, then also errors never get subtracted. So take it like a thumb rule. You never subtract errors. Then life will be much easier. Okay. Anand, have you understood what Aditya said? Yes, sir. He said 2 delta a by a plus 3 by 2 delta b by b plus 5 delta c by c. Okay. This is the correct answer. Okay. I'm not sure. Are you getting it? Yes, sir. Increase the size of the razor. Okay. Let's take a few more questions. We know that resistance is voltage divided by current. Okay. This we have been studying since we were in ninth. So voltage is calculated to be this. 100 plus minus 5 volts. Okay. And the current is calculated to be 10 plus minus 0.2 amperes. Fine. You need to find out the error in the resistance. You need to find the absolute error. You need to find first the relative error. And you need to then tell me the absolute error both. Okay. I thought I lost collection. You're not hearing the music coming from my side. You can use calculators. Okay. Okay. Anybody got the answer? Yes, sir. I got it. Okay. Delta r by r is? 0.07. 0.07. Okay. Are we on both of you got? Yeah. And Aditya? So I got the same answer. 0.07. Okay. Fine. And what is absolute error? Delta r is how much? 0.7. Correct. So it's 0.7. Right. So delta r by r is equal to delta v by v plus delta i by i. Okay. You can write this as v into i to the power minus one. Okay. Now, whether it is power minus one or plus one, it doesn't matter. The export, you need to always add the errors. Okay. So that is why it will be like this. So delta v is five. So five by 100. Plus delta i is 0.2. So 0.2 by 10. So this is seven by 100. And delta r is seven by 100 into R. How you calculate the R? R is just a v by i. So I'll take the actual reading of v and i. Discounting the errors. So R is. 100 by 10, which is 10. So delta R by 10 is equal to 0.07. So delta R is 0.7. Fine. Just one more question, then we'll proceed further. We know that time period of pendulum is 2 pi under root L by G. Okay. Now I'm basically I'm behaving as if I don't know the value of G. Okay. This is uncertain to me. What I'm doing is I'm calculating the length, which is 20 centimeter. Okay. Which is known to me known to one mm of accuracy. Okay. And then I'm calculating the time period of the pendulum. Okay. I'm carrying a time period of pendulum, which is roughly equal to 0.9 seconds. Okay. It is known. It is known to 0.05 seconds of accuracy. Okay. You need to tell me out of these two readings when you have these two readings, you need to tell me what is the error? The relative error in G delta G by G is how much? Have you understood the question? Do it. You want us to calculate it? I want delta G by G. Okay, so I have to get my calculator. You should use calculate. Are you able to proceed or you're stuck? So is it like 1.116 or something like that? The significant figure should be only one. Okay, then one. Relative error is one. Yeah. Relative error is one. Yeah. So 100% error. Others? I'm just calculating. Okay. What is the first step you guys have done? Oh, sorry, sorry, sorry. I missed the decimal point. Both sides bring GS subjective equation and then calculate from there. That will be better. You will get answer from both sides. So it's 0.1. Others? I do it now. How much will be the error in 2 pi? Can you give me that answer? It's zero, right? It's a constant. Correct. It is not a measurement. So error in 2 pi is zero. Any constant has zero error. Yes, sir. Even I got 0.116. Okay. Yeah. So T square is 4 pi square L by G. You asked for delta G by G, right? Yes. Sir, the number that I just got was for delta G. No, what's the delta G by G? I meant delta G by G. So delta G by G, see look at here. So delta G by G is equal to the error in 4 pi square. I'm just using the formula right now. Error in 4 pi square divided by 4 pi square, plus error in L divided by L, plus 2 times error in T divided by T. Now error in 4 pi square is zero. So this goes off. So delta G by G is equal to delta L by L. Now delta L is how much? 0.1. Yeah. I have to convert both should be having same units. It need not be like both should be SI units, but at least they should have same units. So this is 0.1 centimeter. This is 20 centimeters. So 0.1 divided by 20, plus 2 times 0.05 divided by 0.9. Sir, is it 1 by 20 plus 1 by 9? Yeah. This is what it is in front of you. Sir, 1 by 200 plus 1 by 9, right? Yes. How much of that is? That's 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. 0.1. This problem was important because here the subject was time in the equation. But we need to find G. So first we need to rearrange that term, bring G on the left hand side modify the expression and then use delta G by G is equal to 1. You could have got it here as well if you do it here with delta T by T is equal to half of delta delta L by L plus half of delta G by G because the exponent of L and G both is half. So the delta G by G now if you take it on that side it will remain plus only. So then delta G by 2G is delta T by T plus delta L by 2L. Then you multiply by 2 you will get this itself. But why do you do all that circus? First write in terms of G the equation and then modify it. Any doubts till now? Okay, fine. So we are about to complete this chapter. I remember our sound is coming from your side. I am not being safe. Are it is playing music? So my sister is playing music. I asked her to turn off. Sorry. You can always listen to it when you are relaxed. Okay, next the last topic of this chapter best fit lines. Have you ever drawn best fit lines when you have labs? We did an excel graph. What? We did an excel graph. Best fit lines in lab we did an excel graph. Okay. So typically when you suppose you have to verify the ohm flaw. Okay. So what you do is on y axis you take the voltage and on the x axis you take the current and the slope of this graph is your resistance because r is equal to v by i. Have you done this experiment? No. Okay. So we did last year we did it with that variable resistor and we had to vary the resistance in it and then keep measuring the change in current and voltage. Okay. Okay. Understood. So you keep on varying it so voltage and current will vary. So here I am what I am doing is I am calibrating the voltage and current. Okay. So I get different different readings for voltage and current through the same resistance. Okay. To the same resistance I am changing the voltage because of that the current also changes. So I have a meter here which measures the current and I have voltmeter which measures the voltage across the resistance. Okay. So every time I change my source here the potential difference across resistance will change and current also changes because of that I get multiple readings. Suppose one reading is like this other reading could be like that. Okay. Like this I will get readings and it may not be all along one straight line. If the experiment is perfect it has to be along one straight line because ultimately voltage and current graph should be straight line but since your experiment has errors that is why you don't get the perfect straight line. Okay. But the best fit line will be a line that I mean it is an approximation best fit line. There is no direct formula for it. So what we do is that we we know that it has to be a straight line. So we plot it such that it represents the error both sides. Okay. I cannot say that this is my best fit line. You understand. So this is definitely not a best fit line and this one is the best one. Okay. Now suppose you have to find the error in the slope because here the slope represents what the slope represents the resistance. Okay. So the error in the slope will represent the error in the resistance itself. Right. So what I will do is that I will plot the maximum possible slope like this and the minimum possible slope based on the readings I have. Let's say this is a minimum slope. Okay. So if I know this is theta one and let's say this is theta two. How can I write slope in terms of theta one? Do you guys know it? Do you know that tan of theta is the slope? Do you know this or not? So why tan of theta shouldn't be like the sign of theta or the positive theta? No, no, no, no. Just tan. Sir, because tan gives the opposite over education. Yeah, definitely. If you draw a right angle triangle like this, this is your delta Y and this is your delta X. Delta Y by delta X should give you slope and this is angle and this is perpendicular divided by base which is tan theta. Okay. But anyways, suppose you don't know. Okay. So what I what you do is that you find out the maximum gradient. The slope is also called gradient. Okay. So you find out the maximum gradient. Let's say r max is the maximum slope and you find out the minimum slope which is r minimum. You subtract these two and divide them by two. Okay. This will represent this will represent the error in R. Okay. So once you find this delta R, now you can write your reading as R plus minus delta R. Okay. So this is how you deal with the error in the slope. Have you understood? Okay. I'm assuming you guys have understood and tell me one thing. You have done all the questions from this chapter in your textbook. No, not all. Yes, sir. My class did. So the first time we did the chapter, she just made us do all the questions. Aditya, what about you? Yes, sir. Our class also did this theta R, a change in R is equal to r max by r min by 2. I'm asking you, have you done all the exercises from this chapter? No, sir, like we did some in class, but I've not done all. Okay. So I'm assuming that you guys will do these exercises completed yourself and I'll give you material beyond this textbook. Okay. So make sure, since we are done with this chapter, I'll send you assignment from my side, but one more answered assignment is completing all the questions from this book. It may be done in your school. If it is not done, please do it yourself. Okay. But then we have to complete it. All right, guys. So this chapter is over now. So I hope you have understood everything and I'll send you an assignment also so that you properly grasp everything. Now the next chapter is what you guys know? Yes, sir. Well, sir, vectors and scalars is part of the first chapter. Yeah, but then I'm treating it as a different chapter because they're completely my concept. Right, right. That's true. Vectors and scalars is the next chapter. So we did vectors and scalars in like 9th and 10th also. So you can just like brush through them very quickly. You don't need any like, if this is okay with everyone, you can run through this very quickly. Okay. What I'll do is that I'll start, depending on your response, I will increase my speed. And vectors and scalars, you have done in mathematics or physics last class? Both. Math. Mostly math. Mostly math. But we did in physics as well. Okay. You did in physics as well. Okay. Good. So that will increase my speed itself. And see the thing is that whatever we have done till now, measurement and the errors, uncertainty, vectors and scalars, this is mostly mathematical topic. In fact, the vectors and scalars is a mathematical tool. Physics has not started yet. Physics is from the next chapter onwards. Okay. But then all these things are used in physics. So we must understand how it is used in physics rather than just what it is. Okay. So let me just pose a question to you. Can you quickly tell me a few vector quantities in physics? Velocity and acceleration models. Okay. So you know that there are a lot of vector quantities, velocity, acceleration, force, torque, angular velocity, angular acceleration, impulse, momentum, all these are vector quantities. Now, why do I need vector quantities in physics? Should I give, should I talk about the more basic thing? So vectors is magnitude plus direction. Scalars is just magnitude. The scalars are values. Vectors tell you like, like path of movement or like, you know, you know, different things like for energy, it tells you different things, but like it gives you direction also. See till class 11th, till class, let's say 11th, suppose you have not learned about vectors, we dealt only with scalars. Yes or no? As in, we never got introduced to a concept of vectors. Okay. We have been dealing with magnitude of every physical quantity, whether it is mass, energy, pressure, density, temperature, whatever it is, we, we always take help of numbers to explain everything in physics. Okay. But there are certain things which we cannot explain just by numbers. Okay. So what we do, suppose you have to go, go to your home from let's say school. So I need to, if I just tell you you have, your home is five kilometer away, will you be able to reach your home? No, because that is incomplete information. I should give you a sense of direction also. So I'll be telling you, okay, go straight, take second left, then take first right and keep on walking for two kilometers, then take the third right, something like that I have to tell you. Okay. But the problem is when I tell you like this, I'm describing the directions in words. Right. I'm telling you the magnitude only, but I'm also telling you the direction. But the way I'm telling you the direction is that I'm using the words to explain the direction. Yes or no? So it is same way as if, if I explain, let's say, if I explain the mass of R M and mass of Aditya. So if I have to explain without using numbers, I'll just say, okay, R M is heavier than Aditya and Aditya is heavier than R M. So I'm not using any, let's say, I'm not quantifying the magnitude itself. Okay. So it is like describing the magnitude in words. Similarly, when you tell the directions in words, you're not quantifying the direction. You're just telling, you're explaining the direction. All right. Vectors help us to quantify the direction. Okay. So imagine there are no numbers and you have to talk about the magnitude of every object in terms of words. It will be extremely difficult to communicate. Okay. And suppose you communicated it. But if you do not assign a number to let's say mass, then how will you add two masses or subtract two masses? It becomes completely chaotic. Similarly, vectors are required for us to do the mathematical operations. If I do not have a knowledge of vectors, I will not be able to do operations on directions. What will happen if two directions are multiplied? What will happen to that direction that will come out? What will happen, right? So what will happen to the direction if I add the two directions? Right? So in this chapter, our prime focus is to understand the operations using vectors. We I'm assuming that you guys already know that there are a lot of physical quantities that are vectors and they cannot just explain by using magnitude. Okay. So I when I say that I'm using vectors, I will be keep on adding on subtracting forces, velocities, displacement. So I should know how vectors add up, how vectors subtract, how vectors multiply because for scalars, we have spent 10 or 11 years to understand the scalars. But vectors, you are expected to master in let's say two, three hours itself. Okay. So the natural tendency is that whatever you learn in vectors, you tend to correlate with okay, whatever is happening in scalars, similar thing should happen in vectors also. But then you have to keep on reminding to yourself that these two things are completely different from each other. Okay. So the way two numbers are added, the two masses are added are not the same as if two directions are getting added. Okay. So you need to be very cautious and careful. Now scalars are represented by let's say number like this. Suppose four is there, or 36 is there. Okay. So this gives a complete information about scalar because scalar is all about magnitude. But what about direction, what about vectors, how you will represent vectors? Anyone? Now you can quickly reply to me so that I'll keep on moving faster and faster if you have done it already. How will you write about vectors? How do you represent vector? This is the way you represent scalars. So there should be a bar on top of it or an arrow at least. I have to represent both magnitude and the direction. Right. It should have both magnitude and direction. So what about an arrow? I can tell the length correspond to the magnitude and the head correspond to the direction. Is representing like arrow fine? If it is fine, why not a cone like this? I can say that the length of the cone is magnitude and the tip of the cone shows the direction. Okay. An angle relative to the x-axis would be better to show the direction. Okay. Right. So you can, but then if you tell the angle, you're also like describing the direction in words. Okay. Now I can use different notations to tell the direction like what you said angle and geometrically I can take the help of an arrow then I can draw a cone also. Okay. There is nothing wrong in using cone, but this is considered to be the simplest notation. So that is why it is universally accepted that the arrow is the best representation of the vectors. Okay. Now tell me one thing. What will happen to the arrow? Sorry. What I was saying was the arrow represents the vector. Now if I move the arrow parallel to itself, suppose I have moved the same arrow parallel here. Is the length changed? Has the length changed? No. Is the direction changed? No. So vector has both. Sorry sir. There was a little bit of a problem. What happened? I left the meeting by mistake. Oh, I didn't know about that. Anyways, so what I was saying was I hope you are able to understand. So what will happen is that when you move the arrow parallel to itself, its length is not changing and it is still pointing on the right hand side direction. Right. Even the direction is not changing. So can I say that vector remained unchanged if I move it like this? Yes sir. Okay. So when you move the vector parallel to itself, the vector remains same. Okay. So one thing that we have clearly learned here is that I can move the vector parallel to itself. Okay. Now this is a geometrical notation of a vector. Geometry will play a major role when you talk about vectors because it has direction sense. So algebraically also we should know how a vector is represented. Now a variable in when we talk about scalar can be written as X or Y or A or B. So these are the variables which represents scalar. Okay. Now when you put a bar on top of it X bar or Y bar like this when you say this represents vector. Okay. And in your textbook, what they have done is that whenever they're talking about vectors, they're putting the letter in bold. So suppose if I write X like this in bold, then this is this represents vector in your textbook rather than writing bar. You can also bold it. Okay. All you have to do is to let the reader know that it is not a scalar. So there are different ways of writing the variables of vector. You put a bar on top of it or put the letter in bold. Both ways it is fine. Okay. So I can say that this arrow represents let's say A bar vector. So this arrow also represents A bar because it is just moved parallel to itself. Nothing should be changed. Okay. Now tell me what about this vector? If I reverse the vector like this, I have just taken this arrow and reverse it direction. So what would be the representation of this? If this is A bar, what would be this? Minus A bar. Minus A bar? Aditya, do you know this? Yes, sir. Okay. So this is minus A bar. All right. So remember this, I can switch, I can flip the vector and if one vector is A bar, the other one is minus A bar. Okay. So I can jump to this multiplication. Now there are different kinds of multiplication operation in physics. Okay. So there can be multiplication of a vector. Write down multiplication of vector with a scalar. Okay. Tell me an example when this happens. Tell me an equation or example where we multiply scalar with a vector. If there are no such examples in physics, then there are impulse. Impulse. What is impulse? Sir, force into time. Correct. So force into time is a multiplication of a vector force with a scalar time. Okay. Is there any other example? This is impulse. Can you think of any other example where you multiply scalar with a vector? No. The force is mass time acceleration. Right. So mass is a scalar and A is a vector. All right. So in physics, there are such scenarios that exist where you multiply scalar with a vector. So I should know how a vector will react when you multiply with a scalar. Okay. So suppose A bar is a vector and my scalar is, let's say, lambda. All right. And I'm multiplying these two. It becomes lambda A bar. This is my new vector B bar. Now, can you tell some information about B bar? What, how B bar will be related to A bar? Ariman, explain. Sir, I didn't quite get the question. What I'm saying is, suppose B bar is a multiplication of lambda with A bar. Okay. How A bar and B bar are related, that is what I'm asking. Sir, B bar will have, should have the same direction as A bar, but it will have a different magnitude than A bar. Okay. Others? Well, sir, if lambda is negative, then it won't have the same direction. Correct. That is what I wanted to hear. So basically, please write it down. If lambda is zero, what will happen to B? B will become what? Zero. Okay. That is wrong. Zero is a scalar. Okay. Always remember, when you multiply scalar with a vector, you get a vector. Please write down scalar with a vector. The outcome is a vector. Although you may mean the same thing, but the way you say is wrong. So B bar is basically null vector. It's a zero in the vector. Null vector. Like this, you should say. Understood. So this is B bar if lambda is zero. If lambda is greater than zero, the direction of A and B are same. Okay. And if lambda is less than zero, Arima, now you got it. Direction of B will be what? Reverse. Reverse. Opposite of A. Opposite of that of A. Because it's negative, right? Correct. So and in both the cases, the magnitude of the vector. Null vector is a vector which has no magnitude or direction, right? Yes. Zero is a number which has no magnitude. Null vector talks about both direction as well as magnitude. Now, tell me, do you know how to write the magnitude? Suppose B bar represents the full vector. It represents both magnitude and direction. Suppose I have to represent just magnitude. So how, how should I write it? Do you know what is a representation? You guys don't know. Magnitude. This is how you write. Now you're able to remember that you did in 10th. Yeah. Ariman and Anand, are you able to recall? Yes. Okay. This is how you write the magnitude, okay, for a vector. This represents that you are only talking about magnitude. You don't care about the direction. Okay. So magnitude of B in all the cases will be lambda times magnitude of A. Can I say that magnitude of B will be greater than A all the time? Not necessary. If lambda is less than one, then your A will be greater than B. So if lambda is greater than minus one, but less than one, then magnitude will be less than A. Okay. All right. Now, let's talk about the addition of the vector. We've already done this. Addition of vector. See, I want to proceed in a sequential manner. Okay. So let us do it. As in there will always be some new thing that you will hear. Okay. You might have done it. I know it, but tell me how, I mean, tell me a few examples in physics where you add the two vectors. So it could be any, like any examples when like you're taking relative velocity, right? When let's say like two cars are coming against each other and they hit each other, right? Like the magnitude and the direction will be given by the addition of those vectors. Or it could be anything when you have two forces pulling in opposite directions, you add the two vectors, subtract the two vectors to get it resulted. Correct. Good. So addition of vector is a common operation in physics. All right. Now, of course, you guys might be knowing that you'll be either using triangle law of addition or parallelogram law of addition, but I want to get into the logic behind it. So suppose you have two vectors, let's say this is vector A and there is another vector. For the sake of simplicity, I'm drawing it like this only A and B. Now I want to add A and B. Okay. Do you know the answer already? What is A plus B? Well, the vector, you draw it from the tail. Yeah, you draw, yeah. And you meet it with the head of B and you call it vector C, I guess. And vector C equals to vector A equals B. Now, tell me why it should be like this. It is okay to remember it like a law, but can you just take a few examples and tell me that, okay, this is what it is. And that's why all the vectors should be doing like this. If you take it in the case of someone traveling on a road, this would be a displacement factor. Correct. That is what I wanted to hear. So if someone is going from point one to two, and then going from point two to three, and I'm interested in the total displacement. So logically, I should be adding the two displacement to give me the final displacement. Final displacement is the arrow connecting one to three. So that's why it should always create a triangle. Okay. And one more thing you might notice is that if you're adding the two vectors, you must connect the tail of the second vector with the head of the first vector. Getting it. All right. So these are the basic things you must know. Now, since you guys have already done in the 10th, I am directly giving you a problem to solve. So if you do this, I can quickly jump to the next topic. Suppose this is a pentagon. This is vector A, this is vector B, vector C, D, and this is vector E. Oh, you've done this. Then you keep quiet. Others, I want to know some of these vectors. Zero. Zero. Okay, you have done this. So this is A plus B, A plus B plus C, and this will be A plus B plus C plus D, and E is minus of that. So that is when you add up, it becomes zero. Okay, good. So I can move to parallelogram law, parallelogram law of addition. This was strangle law of addition. Who taught you vectors last year? Mr. Vigilakshmi and Mr. Ajay for physics. Mr. Ajay? In Santam, it was Mr. Rohit. Oh, yeah, in Santam, Mr. Tushas. Tushas and Rohit, sir. Yeah. Oh, so they might have covered everything. And I think you might have solved, let's see, Varma as well. Yeah. Okay, so that sounds nice. But just for the sake of completion, I'll just quickly touch upon each and every single topic. So suppose you have two vectors A and B. They're not connected head to tail, but they're connected tail to tail. And I have to add them up without moving any of the vectors. So what you have to do is create a parallelogram like this out of these two sides. And the diagonal will represent A plus B. You guys know this already? Okay, so this vector you can displace here. So this is same as A. So this will be A plus B. Now, can you tell me where is A minus B? Sir, A minus B is basically A plus minus B, right? Yeah, the subtraction of vectors does not exist. Where it is in the parallelogram, how will you draw? Which two points you will connect for A minus B? Two and four. Where will be the head? The head will be four and tail will be two. So you connect the line from two to four to get vector A minus B. Good. So this will be A minus B. This is B. So this will be minus B. Now minus B and A are connected head to tail. So you can add them like a triangle. So if one of the diagonal in the parallelogram represents A plus B, the other diagonal automatically represents A minus B. So you guys clearly well versed with the sum of the two vectors and the difference in the two vectors. Now let me get into a very important part of the vector, which is called components. What else last year you guys have covered? Have you covered kinematics as well? Yeah, sir. We didn't math to a certain extent. Okay. And laws of motion? Rigid body motion, torque and everything? No, not rigid body motion. Torque, not to a very great bit. I think we started it off a bit, but we didn't go much into torque. Actually, this is a tough manner to even start it for. No problem. All right. So components of a vector is something which we are going to use in almost every physics problem that we'll be solving. Okay, because the like what Aditya said that adding up the two forces, subtracting two forces or adding to displacement, okay, these kind of examples are very, very common in physics. So you when you learn laws of motion chapter, you'll see that most of the time you're adding or subtracting two forces or multiple, you know, like that. So if it is just about two forces, you can, you know, add them using triangle law of addition like this. If this is F1 force and this is F2 force, you can just add them up using triangle law of addition and you'll get the some of the two vectors. Now the problem of adding like this is multiple. The first problem is that, you know, it is okay to represent like a triangle, but if someone asks you what is the magnitude of F1 plus F2, you should have drawn this very, very accurately. The length of F1 should be accurate. Length of F2 should be accurate. This angle should be very accurate when you draw it. Then only this length will represent some of the two vectors. Okay, now this is one problem. The other problem is what if you have multiple forces to add these, I can just for the representation sake, I will draw, let's say five forces like this and somebody asked me that they are the five forces acting on an object, find out the sum of these forces. It becomes very, very problematic. I first add these two vectors, then whatever comes out, I'll add this to that, then that, then this. So it becomes very problematic. All right, so there has to be a better mechanism to deal with some of multiple vectors. Okay, and components of vector just does that. So what it says that if you split this vector into two parts, this vector, you can say that it's a sum of these two vector. This vector, you can say that it is some of these two. This one, you can say that it is some of these two. And like this, you can split the vector. So splitting the vector is taking component. So basically, I have said that this vector, since it is some of these two vector, so I need not draw this now. Then this is also gone because this is some of these two vectors. Then this is also gone. Okay, this is gone. But sir, you're resolving these vectors, right? The yellow line which you just removed, those are the resolutions of the vector. That is more important than the two things which are adding to get, because like you can just use the even trigonometry to find the last side. It should be, I think it'll be tan theta, right? See, Aditya, what I'm saying is that if, I mean, what I'm saying is that these two vector is a sum of these two vector, if you add up, you'll get the actual vector. So I'm saying that adding five actual vector as it is is very difficult. Okay, as it is, you'll be having you to deal with multiple angles and you'll go crazy if you draw geometrically. Okay, so that is why what we do is that we split the vectors like this as shown in your screen. Okay, and then add it. Have you understood now? Suppose this x1, x2. I knew what you were doing before, sir. So I just didn't know why you were like splitting it now, I understood because you want to add it. Yes, yes. So that's how I started, right? To tell you that why we need component because adding the two, adding multiple vectors become difficult and things like that. Okay, suppose this is y1, y2, this is y3, y4 and y5. Okay, so if I add up the vectors, horizontally, I can add all the vectors. So this will become x1 plus x2. Now, if I add x2 and x3, their magnitude will get subtracted, right? Because x3 is in opposite direction of x2. All right, so this is minus x3 plus x4 minus x5. Okay, so this is sum of all the components of vector in horizontal direction. And this represents sum of all the components in the vertical direction. So these vectors in the horizontal direction also, they also have like they can be further broken down into i cap, j cap, right? Yes. No, no, no, no, sorry. I am taking my this axis as x axis and that one the y axis. So I'm coming to that. Okay, just take a pause. What I'm saying is that if I add all the horizontal one, it'll be like this, all the vertical one like that. Okay, and then I can add these two vector. I can apply the triangle law of addition. Okay, since this is 90 degree, this will be under root of sum of all axis square that is base and sum of all y square. Okay, and I can also find out how much angle it makes with the x axis tan of theta will be equal to this divided by that. Okay, have you understood till here why we are splitting so that I can add it like this. Arima Anand. Yes. Okay, now what Aditya said was that can I further split this vector? Okay, I want to let's say I want to split this vector. Okay, in a direction perpendicular to itself and in its own direction. Can I do that? Yes, sir. You can use the icap jcap. icap jcap, forget about that right now. icap jcap are the unit vectors along x and y axis. But that will come later on. My question is much more basic. My question is, can there be a component of vector perpendicular to itself? Well, that component would be zero. Yes. Can there be first answer that yes or no? Sir, it can be but it will be a null. It will be null vector. It is nothing, right? Null means nothing. Because perpendicular to it is like, yeah, it's null. Not. It cannot be. So this is the biggest property of vector. Okay, so if my x axis, if my x axis is like this, x1 will not have any ycap component. Are you getting it? x1 will not have any ycap component because y axis is like this perpendicular to x. Fine. So the biggest property of vector that we are exploiting here is that, please write down, a vector doesn't have any component perpendicular to itself. And you will see it will do wonders when we solve physics problem. Okay. Have you written this? Okay. So you will see that going forward when you will study kinematics and all, if there is an acceleration along y axis, it cannot change velocity along x axis. The reason is that acceleration is perpendicular to the x axis velocity. Okay. So since it is perpendicular to x axis, it has no component along x axis. So it cannot affect anything along x axis. Okay. Similarly, if you apply force along x axis, it will never create acceleration along y axis. All right. So because of this property itself, I'm always taking component of a vector perpendicular directions. Okay. So I could have said that, you know, I could have said, okay, this vector is some of these two vectors, one vector like this, one vector like that. There's nothing wrong with it. Are you getting it? But I'm not choosing to write this vector r as some of, let's say x1 and y1, which are not perpendicular to each other. Because if I do it like this, I will never be able to exploit this property of a vector. Okay. To make sure that I use this property of a vector, I am dividing or taking the component of vector perpendicular to each other. Have you understood now, Aditya? Yes, sir. All right. So I think the time will permit us to up till here only. So what I'll do is that I'll quickly finish up this chapter in the next class and we will start with the actual physics next class itself. Okay. And two things. First one is that whether you do it at school or not, it doesn't matter to me. You need to finish your school, book, textbooks, books, questions from measurement chapter, measurements and error. And then I'm going to send two assignments. First one on the errors and uncertainty and the second one on vectors. All right. So I hope this is not, I mean, is it doable? All of you? Yeah. Okay. So I'll share this. Arima, are you saying something? No, sir, I'm just saying I should be able to finish. We'll do next Monday then definitely. Okay. Fine then. So we'll meet next Monday now. Bye.