Understanding the Basics of Vectors

Hello, how are you all? Today we will learn about Vector Class 11th is a very important topic Vector is an introductory video It was made last year, you can watch it It is on the top of the playlist I mean, it's a little... It's a video made in my childhood, so forgive me. But, there are a lot of basic introductions in it. Okay? And some of you said, make a vector, make a vector. So, we came here. And we will talk about vectors and we will talk about good things. Okay? About vectors... If you don't know the basics then watch that video, I have explained it very nicely. Generally, those who have just come here, Vector is a quantity of physics which has direction. For short, let me explain this. The story is going on in the middle. Like our velocity. So we have all read that velocity is direction with speed. For example, if I tell you that 5 m per second and 5 m per second east are the speed or velocity. We can call the velocity of both of them as the same Why? Because it has direction and magnitude In school, teacher must have told you that The one who has magnitude and direction becomes a vector And the one who has only magnitude becomes a scalar We call such quantity as scalar Like our speed And if there is direction in it It becomes our vector quantity Like velocity here What quantity is it? Vector quantity Basically, I mean, little by little... like this you will go inside and read questions and topics then you will enjoy it, not like that you will tell us in the beginning so slowly watch and understand so the quantities which have direction etc. we will call them vector like displacement in displacement we should always tell from where to where like I said I walked 5 km so this is distance I said I walked 5 km towards north so this is our vector means displacement There are many quantities like this. we have force, force always has direction, you see it is like this, two Newton's force is like this, like we have torque, torque do you remember, moment of force, we studied in class 10, that it is rotating clockwise or anticlockwise, so the quantity in which direction is obtained, that quantity we call vector, you can understand this in a basic way, so many concepts are broken, changes will come in life, the name of change is life. how many things are changing in 10 to 11, slowly changing so let's talk about suppose I have made a vector suppose this is velocity vector for some time this is velocity vector and its velocity value is 5 m per second so its arrow point tells direction, the head of the arrow the head of the arrow tells direction and the length of arrow is magnitude what does it tell? magnitude means its length is 5 and direction is east so we will say velocity is 5 meter per second towards east so the length of arrow is magnitude and the head is direction ok now I will show you why vector is a different chapter Why did you open a new chapter for vector? Why did you open a new chapter? As I told you that I have 2 kg sugar and my younger brother also has 2 kg sugar or let's say I give it this way good 3 kg sugar and my younger brother has 4 kg sugar ok now what will we do with so much sugar we should not eat so much sweet right ok how much will it be combined 7 kg sugar very easy very good very good very good very good now I will tell you that I walked 3 meters east and I walked 4 meters north 3 meters east after that 4 meters north Thank you. So tell me how far I have reached from the starting point I have run 3 meters east and 4 meters north So how far am I from the starting point So you will say sir this far And the answer will be 5 Everyone knows Pythagoras theorem So, we have to subtract 3 from 4 and the answer is 5. So, what kind of subtraction is this? This is the subtraction of vector. Because, what is 3 meters east? Displacement. What is 4 meters north? Displacement. and the distance between the initial and final point is also called as Displace so how many are coming after 3 and 4? 5 very good ok I said I went 3 meters east and then 4 meters west 3 meters east and then 4 meters west so how far am I from the starting point? first came forward then went backward how far am I? One meter. Brother, is it three meters ahead or four meters behind? How far is it from the starting point? One meter. Meaning, the answer is coming in combination of three and four. This is the story of the new generation. As I say, three meters ahead and then four meters behind. Now how much will be the answer? 3 plus 4 is 7. Everyone is understanding how far I am from the starting point. This method of 3 plus 4 is 7, 3 plus 4 is 5, 3 plus 4 is 1. This is the story of vector. like we can't add things like sugar, kg, seconds to time because that is scalar quantity to add and reduce vector quantity, to multiply, all this is a different math it is vector algebra, so they have to understand that because vector quantity is to do dv, on which is your whole 11th based? on motion, velocity, displacement, force, moment of force means vector quantity will come on vector So it is very important to understand the vector. So we will tell you how it was. You told 3, 4, 5. You said 3 goes here, 4 goes here. I asked you to tell the answer. You said 5. So 5 comes after 3 and 4. You thought this. It is not like that. Like I said I went 3 meters ahead. And from here 4 meters ahead at 120 degrees. Now tell me how far am I from the starting point? Now tell me is it 5? No right? Here it is 5 that PNP degree Pythagoras Here it is not 5, it looks more than 5 So for all this you will have to learn Today's topic is Vector Addition How do we add vector quantity, force, displacement, philosophy, exploration, exploration? We call it vector addition. I will tell you the three major methods of vector addition. The first is the head tail method. The second is the parallelogram method. and third is our triangle you can say phthalogram law or head tail law so these are the three basic methods if you have studied in good city or good coaching then you can teach this rest these are new and you have to understand and these are similar don't worry so these are the three methods first head tail method is taught to small kids tail method see what is tail method what it tells us suppose you have 1,2,3,4,10 vectors whatever you have to join what you have to do join every next vector with head of previous vector I will write it like this Join tail of next vector with head of previous vector and keep doing this work and when you want the answer and in this vector the name of the answer is resultant We call the answer as Resulted As we said we know by combining A and B So the answer which will be combined with A and B We call it as R Resulted Similarly in this chapter We named it Resulted We called the thing which is made by combining the vector as Resulted and on this we will write something like this this tells us that this is a quantity vector like on any quantity this small arrow means that what is this thing we will write this quickly we will start writing like this if on any thing arrow is placed like this then we will understand that this is a vector and the answer that comes after joining of vectors is also given ok let's come back head tail gadget is join tail of next vector with head of previous vector and continue this 2,3,4 times When we want resultant, we want tail of first vector with head of last vector. Head of last vector. This is the method. Let's take the same example. 3 m east displacement plus 4 m north displacement. Now tell us what is the resultant of this. So, I have made the first vector 3 meters east. Now, what is the rule? Join tail of next vector with head of previous vector. So, its head is here. From here, join the 4 meter tail with the previous vector. Tail of next vector with head of previous vector. Done. Now what we need? Result Join tail of first vector with head of last vector This is result And its value is 5 That is just coincidence We have got 5 answers from coincidence So this is tail law Let's do one more question Ok, note is written Take a copy first It will be a letter for you I don't have any common medical answer copy write on walls write on ground write somewhere write on heart you are writing something else on heart you have come in 11th grade so as we take an example suppose we have to add 3 vectors ok I will give you try to pause and start from your side add 3 vectors suppose I have 3 forces one force is of 3 newton vest One force is of 3 Newton towards North One force is of 3 Newton towards East Now you will tell the answer by combining these three Ok, let's start 3 Newton West, so from here 3 Newton goes to West Where should we go? West. East, West, North, South. I hope everyone knows. Ok, 3 Newton goes in which direction? West. Now, 3 Newton goes in North. So, the tail of the second vector is joined to the head of the first vector. So, 3 Newton goes in which direction? North. Now we have to add third one also, 3N East, sorry 3N, A, B, C. The answer that we have decided is vector A plus vector B plus vector C and all of them are equal to 0. Now third one is 3N East, now with whom we will add it? Previous vector C. Connect all new vectors with previous vector from here to here connect tail of previous vector with previous head 3N Now we need answer For answer we said tail of first with head of last resultant will be like this tail of first join with head of last this is resultant you can see square is being made this is 90 degree and this is 90 degree so definitely its value will be here so 3 3 3 how much answer we got 3 this is miracle is my thing clear Sumatimi Adi It is shining. Very good. So this is a head tail Lata but it cannot be used much. Because there is nothing specific about the angle. For example, if I say that we walk 5 meters in the east, then 5 meters at 60 degree from east Now tell me the answer First understand 5 meters from east, the ball is there Next vector is 5 meters again At which? 60 degree from east This is the east direction Next vector is 60 degree above Sir how do we know above and below? If the angle is negative, then how will we know below? positive is anti-glockwise so 60 degree up means the next vector will be how much higher than east? from here which direction is east? how much higher than east? 60 degree How much is above? 30 And what is the value? 5 Now apply head tail Connect the first tail with the last head Now do you know the answer? No Because it is not looking Pythagoras It is not looking simple geometry So it failed That's why head tail method took a lot of time We have come to the most beautiful, most beautiful, most accepted, most accepted parallelogram law If you want to read a topic, then you tell me. why do you do sarcasm you didn't get this topic done you left it where is it you tell me that you didn't get this topic done we will also do it we should all have positive spirit if someone is offended then I will apologize so parallelogram law this teaches us to connect two vectors in any way without any tension come let's see First we will learn about Law Law says Join two vectors from tail to tail Now understand Now forget about right tail I am not saying literally Forget tail, forget everything, you won't get any answer. You won't get any question. You just have to read it in the syllabus. Join two vectors from tail to tail as two adjacent sides of a parallelogram. Let's take the same example We have the first vector A, whose value is 5 The second vector is B, whose value is 6 And 5 is in East and 6 is 60 degree from East So join two vectors from tail to tail First I made 5 This is the 5th vector. This is the 5th vector. The second vector is B. It is of 6 units. It is 60 degree from east. We don't have to make it from here. Why? Because it is written from tail to tail. So the second vector is made from here. It is of 6 units. It is 60 degree from east. It is 60 degree from east. It is 60 degree from east. The second vector is of 6 units. This is A. Thank you. Now complete imaginary parallelogram Think of parallelogram in your imagination Now resultant will be diagonal of parallelogram Diagonal can be two, one this and one that Diagonal from parallelogram from common point Where two vectors are joined If you measure this diagonal from scale Then this is the value of resultant Wow, how good it is, great Any angle You just have to measure the diagonal of parallelogram Whatever value it has, it will be able to come What is the meaning of parallelogram? Many times in the exam, you have to write to state in the class of 11th grade you will get the first number paper till then you will keep it in the dictionary that first connect two vectors with tail very good then the answer is will be obtained from the diagonal of parallelogram so first what we have to make complete imaginary parallelogram we have to make it from dot and then make diagonal from that point and the length of this diagonal That will be the answer of vector a plus vector b Ok So your mind has come up with the question Sir we will take the scale in front of us And we will do the information And we will take the protector And we will pull it No we will not do this There is a formula for this We have a formula What does the formula say Formula says r square is equal to a square plus b square plus 2ab cos theta Now we have come to the topic of the day. Now note down and start. r square is equal to a square plus b square plus 2ab cos theta. So a plus b is equal to whole square. Along with that there is a term of cos theta. a square plus b square plus 2ab cos theta. Where theta is the angle between two vectors. Like here the angle is 70 degree. so 5 square plus j square plus 2 into 5 into 6 into cos 60 so cos 60 value is 1 by 2 which is R square what is R square? so under root it listen carefully when you root it, give only positive value of R I will take this value of r which is always positive I will go into more detail So we know the formula of parallelogram law and we understood the law Now I will take you to the question and will confuse you Let's go to the question add two vectors 6 units 8 units at 90 degree We have to add two vectors, one 6 unit and one 8 unit So, we assume that A is 6 and B is 8 And the angle between them is 90 degree Do you want to make a diagram? Make one more I have made one vector A Now, we don't have to make another vector from here We have to join tails from here How many angles is B vector at? At 90 degree From here, we will get the number of degrees. Now, we have made the imaginary parallelogram. Dot, dot, dot, dot, dot, dot, dot, dot, dot, dot. It has become a parallelogram. Now, R resided, which is equal to vector A plus vector B. How will it come? From the diagonal. So, this is R. R's value is R squared is equal to A squared plus B squared plus 2AB cos theta. A squared is 6 squared. R squared is 2. 6 square plus 8 square and cos 90 is 0 so this term is 0 6 square plus 8 square is 10 square 36 and 64 so it is 100 so r square is equal to 10 square so you will take r value 10 not minus 10 this is only way to find magnitude what is the way to find magnitude whether I write r square or mod r square more darkness mod means only positive so, very good, very nice R is 10, we are also very happy tell me one thing, what is R? scalar or vector? if you add two vectors, the answer is vector if you add two vectors, the answer is vector so, R is a vector very good, what did you tell us in R? magnitude, we are very happy This is the direction of R If it is in the parallelogram and it is diagonal then it will be at 40 degree Will it be at 45 degree? Sure? It will be at 45 degree, think about it Why? Why will it be at 45 degree? There is no guarantee of being at 45 degree Now here comes an important point that this R is a vector in itself So, when we have a vector, then there will be two characteristics of resultant. One is magnitude of resultant, which you have learnt to calculate. And the direction of resultant. You don't know about this by looking at the figure. You don't have much time to make the figure and measure the angle. So, how to calculate the direction of R? Let's learn this also. Okay? So, as we know there are two vectors, one is vector A and one is vector B. I will prove all the combinations now. And the angle between them is theta. We have made a valogram. And this is the resultant R. So, we give the direction of R from vector A. From where do we give the direction of R? From vector A. R is slightly higher than A How high is it? I have said that alpha angle is higher I have understood How high is it? Alpha angle is higher If I know this alpha Then the direction of R will be fixed How high will be A? Alpha will be higher And the value is known So the direction of R is from vector A to alpha from vector A to angle so the formula will be tan of alpha is equal to B sin theta upon A plus B cos theta ok we will note it and remember it 11th is started Paul said that it will be a great day in life you will have the spring of physics now it has started ok it is fun it is fun and good light weight is good mic will also come in few days it will be good and beautiful ok So tan alpha is equal to b sin theta upon a plus b plus theta. Why do we take tan alpha? Why do we take tan alpha? To take direction. Direction of residue. We know the magnitude of the result. Let me tell you the formula of magnitude. I will write it again. r square is equal to a square plus b square plus 2ab cos theta. Very good. And what is the direction obtained from tan alpha? The formula is b sin theta upon a plus b cos theta. Let me explain you one thing. Let's assume that we will not know the direction from a. We will know it from b. we will not tell the direction with A, we have two vectors we have to tell the answer, we know the answer B value now the direction of the answer is to tell the angle with any of these vectors so you told with A, so this formula is made if you tell with B, so what angle will you use? you will use beta, so tan beta tell yourself what will happen, tell tell you have studied so much math turn it upside down, tan beta What will be the equation of theta? a sin theta upon b plus a cos theta We don't have to rectify that we can only tell from a No, we can tell from b What will be the formula of tan theta? We will tell from the formula a sin theta upon b plus a cos theta if you keep a formula then your brain will get confused you are a small kid so keep a formula in your mind then you will apply the converse formula then you can tell the angle from B vector or if you want to tell the angle from B then you have to use your brain if you know theta and alpha then you have to find So, how to solve the equation of 10 alpha is equal to b sin theta upon a plus b cos theta? So, how to solve the equation of 10 alpha is equal to b sin theta upon a plus b cos theta? Let us derive the equation of 10 alpha is equal to b sin theta upon a plus b cos theta. both formulas first we will do derivation r square is equal to a square plus b square plus 2ab cos theta because we don't just teach for y,t,j and nil we teach all the basics of 11 so derivation is very important come let's understand how to do it so we will use law in this we will assume that we have two vectors vector a vector b and between them is angle theta and the resultant is r these things we have given make a,b as you want I will make it a little beautiful beautiful all my dear children I made vector A and vector B and vector B I made it as I wanted it can be made here or here it can be made anywhere this is angle theta remember one thing theta join it tail to tail But you will be very clear about the questions that will come next. What did you say? Take theta angle only when both vectors are connected from tail to tail. Okay, now make imaginary parallelogram. Imaginary... It is done. This is resulted. R Now I want R's value. And I have to get this formula. I am doing one thing, I am extending it further. I have mixed it up and name this parallelogram PQ and we have already changed the R let's do this MNOP and what I did is I dropped a perpendicular from O here I have dropped a perpendicular from O point and called it as cube Do you understand? I have made a parallelogram I have dropped a perpendicular from O point and called it as cube suppose we have a parallelogram, what is a parallelogram? the pair of opposite sides of parallel and equal means if this is B then this will also be B and if this is A then this will also be A means if this side is B then this side will also be A the length of this side will be equal to the length of B vector so we can call this B ok ok, now see from here this angle is theta I will show here if this angle is theta then this angle will also be theta corresponding angle this is theta and this is also theta because both sides are parallel and this is the same line so I can say that this angle will also be theta Very good. Now see a small triangle. Small, beautiful, small, small, P, O, Q triangle. Can you see clearly? Now I will ask you that what is the value of PQ? How will you tell? We will use our brain. What is the value of cos theta, guys? cos theta is base upon hypotenuse what is base of this triangle? pq what is hypotenuse? po cos theta is equal to base which is pq upon hypotenuse which is po from here pq will be po into cos theta and how much is po? b so I got b cos theta which means this small length pq PQ which is small in length will be our P cos theta How to find it? cos theta is equal to base upon hypotenuse Base upon hypotenuse, base is equal to P cos theta Ok, clear? Then I said I also have to find OQ I also have to find OQ So I said in this same triangle, within the same triangle You tell me what will be the value of sin theta? What is sin theta? perpendicular to hypertenes so perpendicular to what? oq and how much is hypertenes? po perpendicular to? hypertenes ok so I don't know the value of oq and what is the value of po? b is equal to sin theta so what is the value of oq? b sin theta so what will be this value? b sin theta I can see fear in your eyes To make the poster fall is my goal This is the goal of my life To make the poster fall The children who are scared and distressed I will not let them be scared I will make it simple These two maths can be taught by anyone How to catch it in a short time Suppose you see a vector tomorrow B and this angle is theta listen carefully if I have dropped this perpendicular then the one below will always be B cos theta and this one will be B sin theta we call this as component ok, we will come to this later I have explained it just now if this is B and this angle is theta then B cos theta and B sin theta I will explain it a little more like suppose Listen carefully, this is B and this angle is theta. Now listen carefully, the angle theta is perpendicular to that angle, and this distance will be B cos theta, and the perpendicular distance will be B sin theta. That means, the angle theta is on the base, B cos theta, and the angle perpendicular is B sin theta. where angle is theta, there is cos theta and in front of it is sin theta where angle is theta, which angle is theta, on this line, how much will it be? b cos theta and how much will it be on its perpendicular? b sin theta suppose this is r, r is making theta angle from this line, so how much will it be here? r cos theta and how much will it be on its perpendicular line? r's component, r sin theta, means this This one, R sin theta and base value R cos theta. Remember this. You don't need to tell this in the exam. If you want to say straight, this is V, then say V cos theta. If you want to say straight, this is V sin theta. We will derive the main formula. You have to pay attention. Now you have to take triangle OQM This triangle is at 90 degree So this is right angle triangle We can apply Pythagorean theorem on this Now we will say MO square is equal to perpendicular OQ square plus base MQ square Can we apply or not? Now what is the value of MO? are value. Here we are talking about value only, what is the value of the diagonal? So, what is the value of MO? R square. What is the value of OQ? B sin theta. So, B sin theta whole square. and how much is MQ? from here to here what is the value? A and this is B cos theta so how much is MQ? A plus B cos theta squared come so from here it will be B square sin square theta expand it a square plus b square cos square theta a square plus b square plus 2ab cos theta this is coming from this this is coming from b this is the whole square expansion of a plus b a square plus b square plus 2ab and here comes cos theta so do one thing from this term and from this term take b square as common So what will be left? Si square theta plus cos square theta. so 1 will be equal to, here I am doing it, first write a square then b square is taken as common so what is left? sin square theta plus cos square theta and here I have 2ab cos theta now sin square theta plus cos square theta, how much will be my? 1 will be done so a square plus b square and what is this? 2ab cos theta, whose equal is it? I think you understood it very well that we have taken this triangle, the value of R is the square of the aperture which is equal to the square of the perpendicular plus the base is the square of the aperture we have expanded it by a square plus b square and from this we have taken b square commonly and derived the formula. Very good. Come, let us derive the second formula. in which I had to tell that the resultant has different angles where is the resultant? Direction so, our next formula is Direction of resultant The ocean will fill with drops Slowly slowly you will understand the physics 10 alpha formula b sin theta upon a plus b cos theta Come let's make the same diagram Here a is made Here b is made Imaginary parallelogram is made This angle is theta, from here we have made a resultant r This is called point M, this is called point N, this is called point O, this is called point P From here we will make a perpendicular on this line This is called point Q, this is point B, so this is also point B This angle is theta, so this angle is also theta Now tell me one thing, where b makes theta angle, how much value comes on that side? b cos theta, means its value will be b cos theta and the value of the perpendicular line of this line will be b sin theta Now see the triangle OQM Now this is the resultant And we know the angle from A to A Resultant is a vector and the direction of the vector is given by angle How much angle is it making from the vector A? How much angle is it making from the vector A? Alpha Now look at this angle OQM Can we get tan alpha value? What is tan alpha in this angle? Perpendicular upon base Look at this angle, who is the perpendicular? I think it is OQ Who is the base? Thank you. and that will be because it will be of value b sin theta and mq value is a and b cos theta so a plus b cos theta is m so these were two small derivations which are asked in the level and after derivation you feel that you have got it so remember to make a diagonal and extend it so if you have got it then it is good if you have not got it then you should repeat it So kids, we have done the derivation. Now, first of all, Let's ask some questions. We will ask some good questions and some easy questions. Let's ask some easy questions first. So, he told us, two forces of magnitude 6 N each act at a point. as shown find the resulting and we have to find the magnitude and direction of the resultant so I will tell you that this is the point and here is a force of 6 Newton and here is a force of 6 Newton and here is an angle of 60 degree now you have to find the resultant of this we have to find the resultant of this and I will give you the option and your option will be 6 root 3 at 30 degree 3 root 3 at 30 degree 2 root 3 at 45 degree Option D we will take 6 at 45 degree Come lets solve this So guys, the result will be from here. If you make a diagram in the beginning, you will enjoy it. The result will be R square is equal to A square plus B square plus 2 AB cos 3. We can write it very easily. A's value is 6 square plus 6 square plus 2 into 6 into 6 into cos 60. Very good. 36 plus 36 plus, how much will this be? 2 into 6 into 6 and cos 60's value is 1 by 2. So here comes the do. So, I can say 6 square 6 square 3 times. So, 3 multiplied by 6 square. Let's write 6 square 6 square 6 square. R square value. From here, what will be the value of R? We will under root it. Under root of 3 into 6 square. So, 6 times square means 6 will come out. So, what will be left? Root 3. So, what will be the resultant value? 6 root 3. There is only one option where 6 root 3 is there. But still, for the satisfaction of our heart, we will also take out this angle alpha. So, tan alpha is equal to what? together. He's 90 da. upon a plus b cos theta now sir which is a from this see a is that from which angle is given means this one although value is same as well so 6 into angle is 60 upon 6 plus 6 cos 60, this time a is also 6 and b is also 6 so sin 60 is root 3 divided by 2 divided by 6 plus 6 cos 60 is 1 by 2 so it is 23006, above 3 is root 3 below 23006 and 3 is 9 so it will be root 3 divided by 3 How to rationalize this? What can I write as 3? Root 3 into root 3 Can you see? Root 3 into root 3 So how much is it? 1 by root 3 So tan alpha is 1 by root 3 So you know when tan alpha is 1 by root 3 When alpha is 30 degree So the answer matches When does tan alpha happen by root 3? When tan alpha is 1 by root 3 So alpha is 30 degree If tan alpha is root 3, then alpha is 60 degree If tan alpha is 1, then alpha is 45 degree and we have to remember that time will come with time and it is good that it is correct you must have understood the example here I would like to tell you one special thing if both vectors are same in magnitude then the resultant will go from the middle it will bisect If the magnitude of both vectors is same then resultant will bisect the angle See how much angle the resultant came at? 30 degree If you want you can memorize this resultant, it is very easy in exam That there are two vectors, one vector is x and other vector is x How much is the reach of angle? 60 degree then the answer is always X root 3 and angle is 30 degree if two vectors are equal at 60 degree then the answer will be X root 3 and this is very common thing and there is no big thing to know about it so let's take more questions the lack of light in our company I am not going to use generator or else the light will be off I am just reading the question Let's take another question Let's try one more question Two vectors of equal magnitude Let's try this one of equal magnitude are added to give resultant which is of same magnitude as the two vectors. Find the angle between two vectors. You have to tell me what angle is between those two vectors whose resultant is equal to those two vectors Means I have one vector A, one vector B and this is our resultant R What happened is that the resultant value is equal to A and B and A and B are equal to each other. What is this? Two vectors of equal magnitude R added to give resultant which is of same magnitude resultant magnitude is also same so we are asked that what is the angle between these two vectors find the angle between two vectors their angle is theta so all these are magnitude and equal direction is not possible let's calculate it R square is equal to a square plus e square plus 2ab cos theta So, in the place of r, a, b, consider all as x So, r x square x square plus x square plus 2 into x into x cos theta So, x square x square, how much is this? 2x square, how much is this? 2x square cos theta We don't know theta, we have to find out this only Ok, x square x square, look at the other side So, this will be x square minus 2x square, so minus 2 minus x square is equal to 2x square cos theta. Now, I have to get cos theta out of here. So, cos theta is equal to minus x square divided by 2x square. You will cancel x square x square. And, you will cancel it because x has no value of 0. This will be discussed in the exam. If x is 0, then never cancel these three things in the paper. In Maths, if x is 0, then you cannot cancel it. Anyways, here x is 0, so we cancel it. So, the value of positive theta is minus 1 by 2. Maybe you have not seen this value in your class yet. When does positive theta come minus 1 by 2? Slowly slowly you will understand. When theta is 120. When angle is 120, then value of cos theta is minus 1 by 2 For this also we will have a chapter on mathematical cube In which we will tell you how to write angle above 90, above 180 For now you can remember that cos theta is minus 1 by 2 When theta is 120 degree So this was easy, good, just information Keep collecting information in your mind You will get new information everyday So this type of questions will be easy for you I will ask you a good question, get ready It is a big question, you will get the answer from inside Let's go out and make it stand Make it stand Sure? Come on Are you sure you are ready? Or not? Come on We have two wickers When you are making videos and teaching You should be careful To teach like this You should teach basic things It's not fun You should be very careful Two wickers Take any name Whatever you like Take P and Q P and Q As a sum of As a sum of 18 and their resultant is 12. The resultant is perpendicular to smaller of two vectors. Find the... value of p and q and angle between them ok, so we have got a good question are you feeling uncomfortable? no problem, we will be with you, no tension see carefully, there are two vectors p and q whose sum is 18, understand this, their sum is 18 and the resultant of y is 12 both these things are different if we add the value of p plus q then it will be 18 if we add vector p and vector q then the value will be 12 I think you are understanding the meaning of these things suppose we add 10 and 8 so 18 can be the answer, value wise, only the value is added but in vector, it is possible to add 10 and 8, so it is possible to get 12 because the answer in vector depends on the angle so these are the two information that they have given us and we have learned that we have to find the resultant there is one more information the resultant is perpendicular to smaller of two vectors the resultant is perpendicular to the smaller vector now let's make an icon so which is the smaller vector in it? we will assume that p is smaller vector I have made p as smaller vector and resultant is perpendicular to p resultant is perpendicular to p now tell me one thing should I make q here? No, we can't make it like this. Because how do we make resultant? With diagonal. So why we have to make Q here? This is one of my favorite question. So why you make Q here? Now see, is it correct? Now see the chances of getting right At least this looks like a diagonal or not PSA QS Who told you that it can't be like this? Who told you? First it can be like this This, this, this, this Now see this will be a reserquent Now only this is perfect So, we have to keep this approach in mind. If you make Q here, then automatically the mind will move. Very good. Very beautiful. And this is the angle between them. Now, we have to use two information. That P and Q have a connection of 18. The vector sum of p and q, that is the resultant, is bar. So let's apply the resultant formula, I think it will be useful. So what is the resultant formula? r square is equal to a square plus b square plus 2ab cos theta. We don't even know theta, we don't even know r, we don't even know p, we don't even know q. No, we know r, so what will we get from r? What will you get from this in life? p square plus q square plus 2pq, we don't know anything. I'm sorry, you got nothing what is the second formula? angle one you know the angle of this how much is the angle of resultant from P? 90 degree you know the value of alpha how much is alpha? 90 degree from whom? from vector P or you can call it A or B whatever you like now tell me formula of tan alpha B sin alpha sin theta upon a plus b cos theta tan alpha tan 90 tan 90 is equal to b sin theta b is q sin theta divided by a a means p p plus q cos theta tan 90 is equal to 10th class Some will have read it as infinite and some as undefined. The one who has read it... I am ready to respect It is undefined if you will argue with me Ok, this thing is undefined or infinity Whichever you like You can fight in comment Take out rally Whatever This thing is undefined or infinity When the thing below is zero In your calculator do 1 divided by zero It will be written not defined Or it will be written Math error If there is zero in the denominator, then only the undefined infinity comes. This denominator will be zero. We got one very important information that p plus q cos theta will be zero. That's it. This is the information we got. P plus Q is equal to 0. So, can I say that Q is equal to minus P. Q is equal to minus P. I will take this formula. 12 square is equal to P square plus Q square plus 2P. And I can put Q is equal to minus P. Q is equal to minus b It looks like a simple thing. P square plus Q square minus 2P square. So this will be Q square minus P square. Will it be possible? Yes, tell me. Q square minus P square is equal to 12 square. Okay, tell me further. What can I write as Q square minus P square? Look here. I can write Q square minus P square as Q minus P Q plus P. Yes. Yeah. a square minus b square a minus b, a plus b is equal to 12 square that is 144 now they have given the value of q plus p 18 now it is time to use this q minus p into 18 is equal to 144 18, 8 is 144 so we got the value of q minus p 1 well done and the value of q plus p is 18 Question Time I will add both of them 2Q is equal to 26 So what is the value of Q? 13 Q is 13 so P is 5 13 5 18 Q is 13 So P is 5 So what is the value of 13 and 5? 18 Satisfy sir Great question, great method Very good question Will you take out theta also? Yes, we will take out theta also Why we have kept cos theta Minus what? Now let's use that. Let's use that. Q cos theta is equal to minus b. Q cos theta is equal to minus b. How much did Q equalize to? 13. cos theta is equal to minus b. b is equal to 5. cos theta is equal to minus 5 divided by 13. Just leave the answer here. Now you can't remember all angles of 5 by 13 So you have to remember that There is an angle theta whose cos theta is minus 5 by 13 Will we do this in mathematics? If we don't have an emulator in our hands then how many angles will you remember? cos 21, cos 22, cos 27, cos 28 You can't remember all of them So you don't know if there is an angle whose cos theta is minus 5 by 13 You have asked a very good question Let's go Ok, you want to give a homework? Ok, I will give you a similar question in a similar way. I will bring it. Ok, I was watching DC Panday and I got a similar question. This question is very old. DC Panday has a similar question. So, a very similar question. Try it once. See, it is fun. I hope it should be done. Take the name of God. Your own. God will help you. Stay happy, do meditation in the morning Do something in life I am talking to you, find the question And I got the question We have been told that the sum of two forces at a point is 16 Newton The sum of two forces at a point is 16 Newton This is the sum, not the result if The resultant is normal to smaller force and has value, Resulted t value is 8 Newton find two forces Two forces and option given to them 6,10 8,8 4,12 2,14 So, sum is 16. 6, 10, 18, 16, 4, 12, 16, 2, and? 14, B. Now, try it. See how the question will be. Try it yourself. Okay? Do it like this. Make a small vector here, A. And make another vector here, B. And make this resultant R. What will be the angle? 20. And what will be this angle? Theta. Then you will put the formula of tan alpha. Which angle is alpha? This one. Resultant angle is with A. Tan alpha, what will you say from this? B sin. I will help you a little bit 10 alpha is 90 and 10 90 is infinity If this is undefined then the term below will be 0 So a plus b cos theta is equal to 0 b cos theta is equal to minus a R square is equal to a square plus b square plus 2ab we will put the value of r we will put the value of r so we will have r square is equal to a square plus b square plus 2a and b is the cost at this place minus a so it will become b square minus 2 square a square so b square minus a square is equal to 8 so b minus a and whole b plus a is equal to 8 square means 64 so b minus a into b plus a is equal to so now we will solve the first two ok now we will solve it by ourselves try the question yourself a plus b will give you a minus b value a plus b will give you 16 a minus b will give you a minus b value so guys these were the questions on vector addition i hope you liked it and i am thinking that We have time. So I will explain you about subtraction of vector. Time is left. So we will do subtraction of vector today. So I will give you the concept of subtraction of vector. Wait, I will ask you a question. Find resultant. Let me see how many people do it correctly. And all of you pause and find it. Find resultant of two vectors. show ok now we discuss now we pause we have one vector 6 one vector 8 Our vector is 8, angle is 120, and I have told you the value of cos 120, so don't worry. Minus 1 by 2, now you have to find the resultant value. Okay, everyone find it. Pause and find it. I am writing the option here. You will get the option that you have not got it. The question is very important. Very, very important. Okay? So you will get the answer that you have not got it. I will quickly answer it. Ok, tell me I hope you all have asked the same question Ok, now try it So what will be the formula? r square is equal to a square plus b square plus 2ab cos theta You remember the formula, so you can tell 6 square plus 8 square plus 2 into 6 into 8 into cos 120 minus 1 by 2 2 from 2 is dead 6 square 36, 8 square 64 Here minus will be done because it is minus minus 6 to 8 is 48, so 100 minus 48 Answer is 52 r square's value is 52, so r's value under root is 52 Very very good. And the answer is completely wrong. What did we tell you? We told you that theta angle is to be joined from tail to tail. And you ignored this. Here see, the tail of the other vector is joined from the head of this vector. That is head-tail matter. It is not confused. Tail is joined from tail. Tal is joined from tal. Tail is joined from tail. Wrong. You have got a wrong question. You have made this question wrong in its entirety. Come, let's make the diagram again. 6. Now I will put this vector here. 8. Everything is allowed. Parallel shifting of vector is allowed. I can put it from here to here. You can increase this vector and move forward In the first video I taught you how to find the angle between two vectors So watch the previous video of vector Then you will know what is the angle between two vectors And how to find the angle if not given So either increase this or decrease this Now tell me children This angle is 120, this angle is 60, this angle is 60, this angle is 60 Actually, the angle between these two vectors is 60 If you want, you can increase this vector and increase it further This is also allowed, I increased it to parallel Now, what will be this angle? 60 If you didn't get this, then don't cry, watch the previous video, I have told you this with a lot of love and love. Now see, angle is 60, now r square is equal to, yes you are laughing, now anyone will find out sir. 2 into 6 into 8 into cos 60, tell the value of cos 60. Cos 60's value is 1 by 2, 2 multiplied by 2 is 68, 48, this is 100 right. Plus 48, 148, r square, so the value of r and the root is 148. Congratulations if you had hit the C, in the moment you will definitely lie, sir I had hit the C.

But, there are a lot of basic introductions in it. Okay? And some of you said, make a vector, make a vector. So, we came here.

And we will talk about vectors and we will talk about good things. Okay? About vectors...

If you don't know the basics then watch that video, I have explained it very nicely. Generally, those who have just come here, Vector is a quantity of physics which has direction. For short, let me explain this.

The story is going on in the middle. Like our velocity. So we have all read that velocity is direction with speed. For example, if I tell you that 5 m per second and 5 m per second east are the speed or velocity.

We can call the velocity of both of them as the same Why? Because it has direction and magnitude In school, teacher must have told you that The one who has magnitude and direction becomes a vector And the one who has only magnitude becomes a scalar We call such quantity as scalar Like our speed And if there is direction in it It becomes our vector quantity Like velocity here What quantity is it? Vector quantity Basically, I mean, little by little... like this you will go inside and read questions and topics then you will enjoy it, not like that you will tell us in the beginning so slowly watch and understand so the quantities which have direction etc. we will call them vector like displacement in displacement we should always tell from where to where like I said I walked 5 km so this is distance I said I walked 5 km towards north so this is our vector means displacement There are many quantities like this. we have force, force always has direction, you see it is like this, two Newton's force is like this, like we have torque, torque do you remember, moment of force, we studied in class 10, that it is rotating clockwise or anticlockwise, so the quantity in which direction is obtained, that quantity we call vector, you can understand this in a basic way, so many concepts are broken, changes will come in life, the name of change is life.

how many things are changing in 10 to 11, slowly changing so let's talk about suppose I have made a vector suppose this is velocity vector for some time this is velocity vector and its velocity value is 5 m per second so its arrow point tells direction, the head of the arrow the head of the arrow tells direction and the length of arrow is magnitude what does it tell? magnitude means its length is 5 and direction is east so we will say velocity is 5 meter per second towards east so the length of arrow is magnitude and the head is direction ok now I will show you why vector is a different chapter Why did you open a new chapter for vector? Why did you open a new chapter? As I told you that I have 2 kg sugar and my younger brother also has 2 kg sugar or let's say I give it this way good 3 kg sugar and my younger brother has 4 kg sugar ok now what will we do with so much sugar we should not eat so much sweet right ok how much will it be combined 7 kg sugar very easy very good very good very good very good now I will tell you that I walked 3 meters east and I walked 4 meters north 3 meters east after that 4 meters north Thank you.

So tell me how far I have reached from the starting point I have run 3 meters east and 4 meters north So how far am I from the starting point So you will say sir this far And the answer will be 5 Everyone knows Pythagoras theorem So, we have to subtract 3 from 4 and the answer is 5. So, what kind of subtraction is this? This is the subtraction of vector. Because, what is 3 meters east? Displacement.

What is 4 meters north? Displacement. and the distance between the initial and final point is also called as Displace so how many are coming after 3 and 4?

5 very good ok I said I went 3 meters east and then 4 meters west 3 meters east and then 4 meters west so how far am I from the starting point? first came forward then went backward how far am I? One meter.

Brother, is it three meters ahead or four meters behind? How far is it from the starting point? One meter. Meaning, the answer is coming in combination of three and four. This is the story of the new generation.

As I say, three meters ahead and then four meters behind. Now how much will be the answer? 3 plus 4 is 7. Everyone is understanding how far I am from the starting point. This method of 3 plus 4 is 7, 3 plus 4 is 5, 3 plus 4 is 1. This is the story of vector. like we can't add things like sugar, kg, seconds to time because that is scalar quantity to add and reduce vector quantity, to multiply, all this is a different math it is vector algebra, so they have to understand that because vector quantity is to do dv, on which is your whole 11th based?

on motion, velocity, displacement, force, moment of force means vector quantity will come on vector So it is very important to understand the vector. So we will tell you how it was. You told 3, 4, 5. You said 3 goes here, 4 goes here.

I asked you to tell the answer. You said 5. So 5 comes after 3 and 4. You thought this. It is not like that.

Like I said I went 3 meters ahead. And from here 4 meters ahead at 120 degrees. Now tell me how far am I from the starting point?

Now tell me is it 5? No right? Here it is 5 that PNP degree Pythagoras Here it is not 5, it looks more than 5 So for all this you will have to learn Today's topic is Vector Addition How do we add vector quantity, force, displacement, philosophy, exploration, exploration? We call it vector addition.

I will tell you the three major methods of vector addition. The first is the head tail method. The second is the parallelogram method.

and third is our triangle you can say phthalogram law or head tail law so these are the three basic methods if you have studied in good city or good coaching then you can teach this rest these are new and you have to understand and these are similar don't worry so these are the three methods first head tail method is taught to small kids tail method see what is tail method what it tells us suppose you have 1,2,3,4,10 vectors whatever you have to join what you have to do join every next vector with head of previous vector I will write it like this Join tail of next vector with head of previous vector and keep doing this work and when you want the answer and in this vector the name of the answer is resultant We call the answer as Resulted As we said we know by combining A and B So the answer which will be combined with A and B We call it as R Resulted Similarly in this chapter We named it Resulted We called the thing which is made by combining the vector as Resulted and on this we will write something like this this tells us that this is a quantity vector like on any quantity this small arrow means that what is this thing we will write this quickly we will start writing like this if on any thing arrow is placed like this then we will understand that this is a vector and the answer that comes after joining of vectors is also given ok let's come back head tail gadget is join tail of next vector with head of previous vector and continue this 2,3,4 times When we want resultant, we want tail of first vector with head of last vector. Head of last vector. This is the method.

Let's take the same example. 3 m east displacement plus 4 m north displacement. Now tell us what is the resultant of this.

So, I have made the first vector 3 meters east. Now, what is the rule? Join tail of next vector with head of previous vector.

So, its head is here. From here, join the 4 meter tail with the previous vector. Tail of next vector with head of previous vector. Done.

Now what we need? Result Join tail of first vector with head of last vector This is result And its value is 5 That is just coincidence We have got 5 answers from coincidence So this is tail law Let's do one more question Ok, note is written Take a copy first It will be a letter for you I don't have any common medical answer copy write on walls write on ground write somewhere write on heart you are writing something else on heart you have come in 11th grade so as we take an example suppose we have to add 3 vectors ok I will give you try to pause and start from your side add 3 vectors suppose I have 3 forces one force is of 3 newton vest One force is of 3 Newton towards North One force is of 3 Newton towards East Now you will tell the answer by combining these three Ok, let's start 3 Newton West, so from here 3 Newton goes to West Where should we go? West. East, West, North, South.

I hope everyone knows. Ok, 3 Newton goes in which direction? West.

Now, 3 Newton goes in North. So, the tail of the second vector is joined to the head of the first vector. So, 3 Newton goes in which direction?

North. Now we have to add third one also, 3N East, sorry 3N, A, B, C. The answer that we have decided is vector A plus vector B plus vector C and all of them are equal to 0. Now third one is 3N East, now with whom we will add it? Previous vector C.

Connect all new vectors with previous vector from here to here connect tail of previous vector with previous head 3N Now we need answer For answer we said tail of first with head of last resultant will be like this tail of first join with head of last this is resultant you can see square is being made this is 90 degree and this is 90 degree so definitely its value will be here so 3 3 3 how much answer we got 3 this is miracle is my thing clear Sumatimi Adi It is shining. Very good. So this is a head tail Lata but it cannot be used much. Because there is nothing specific about the angle. For example, if I say that we walk 5 meters in the east, then 5 meters at 60 degree from east Now tell me the answer First understand 5 meters from east, the ball is there Next vector is 5 meters again At which?

60 degree from east This is the east direction Next vector is 60 degree above Sir how do we know above and below? If the angle is negative, then how will we know below? positive is anti-glockwise so 60 degree up means the next vector will be how much higher than east? from here which direction is east? how much higher than east?

60 degree How much is above? 30 And what is the value? 5 Now apply head tail Connect the first tail with the last head Now do you know the answer? No Because it is not looking Pythagoras It is not looking simple geometry So it failed That's why head tail method took a lot of time We have come to the most beautiful, most beautiful, most accepted, most accepted parallelogram law If you want to read a topic, then you tell me. why do you do sarcasm you didn't get this topic done you left it where is it you tell me that you didn't get this topic done we will also do it we should all have positive spirit if someone is offended then I will apologize so parallelogram law this teaches us to connect two vectors in any way without any tension come let's see First we will learn about Law Law says Join two vectors from tail to tail Now understand Now forget about right tail I am not saying literally Forget tail, forget everything, you won't get any answer. You won't get any question.

You just have to read it in the syllabus. Join two vectors from tail to tail as two adjacent sides of a parallelogram. Let's take the same example We have the first vector A, whose value is 5 The second vector is B, whose value is 6 And 5 is in East and 6 is 60 degree from East So join two vectors from tail to tail First I made 5 This is the 5th vector. This is the 5th vector. The second vector is B.

It is of 6 units. It is 60 degree from east. We don't have to make it from here.

Why? Because it is written from tail to tail. So the second vector is made from here.

It is of 6 units. It is 60 degree from east. It is 60 degree from east.

It is 60 degree from east. The second vector is of 6 units. This is A.

Thank you. Now complete imaginary parallelogram Think of parallelogram in your imagination Now resultant will be diagonal of parallelogram Diagonal can be two, one this and one that Diagonal from parallelogram from common point Where two vectors are joined If you measure this diagonal from scale Then this is the value of resultant Wow, how good it is, great Any angle You just have to measure the diagonal of parallelogram Whatever value it has, it will be able to come What is the meaning of parallelogram? Many times in the exam, you have to write to state in the class of 11th grade you will get the first number paper till then you will keep it in the dictionary that first connect two vectors with tail very good then the answer is will be obtained from the diagonal of parallelogram so first what we have to make complete imaginary parallelogram we have to make it from dot and then make diagonal from that point and the length of this diagonal That will be the answer of vector a plus vector b Ok So your mind has come up with the question Sir we will take the scale in front of us And we will do the information And we will take the protector And we will pull it No we will not do this There is a formula for this We have a formula What does the formula say Formula says r square is equal to a square plus b square plus 2ab cos theta Now we have come to the topic of the day. Now note down and start. r square is equal to a square plus b square plus 2ab cos theta.

So a plus b is equal to whole square. Along with that there is a term of cos theta. a square plus b square plus 2ab cos theta. Where theta is the angle between two vectors. Like here the angle is 70 degree.

so 5 square plus j square plus 2 into 5 into 6 into cos 60 so cos 60 value is 1 by 2 which is R square what is R square? so under root it listen carefully when you root it, give only positive value of R I will take this value of r which is always positive I will go into more detail So we know the formula of parallelogram law and we understood the law Now I will take you to the question and will confuse you Let's go to the question add two vectors 6 units 8 units at 90 degree We have to add two vectors, one 6 unit and one 8 unit So, we assume that A is 6 and B is 8 And the angle between them is 90 degree Do you want to make a diagram? Make one more I have made one vector A Now, we don't have to make another vector from here We have to join tails from here How many angles is B vector at?

At 90 degree From here, we will get the number of degrees. Now, we have made the imaginary parallelogram. Dot, dot, dot, dot, dot, dot, dot, dot, dot, dot.

It has become a parallelogram. Now, R resided, which is equal to vector A plus vector B. How will it come? From the diagonal.

So, this is R. R's value is R squared is equal to A squared plus B squared plus 2AB cos theta. A squared is 6 squared.

R squared is 2. 6 square plus 8 square and cos 90 is 0 so this term is 0 6 square plus 8 square is 10 square 36 and 64 so it is 100 so r square is equal to 10 square so you will take r value 10 not minus 10 this is only way to find magnitude what is the way to find magnitude whether I write r square or mod r square more darkness mod means only positive so, very good, very nice R is 10, we are also very happy tell me one thing, what is R? scalar or vector? if you add two vectors, the answer is vector if you add two vectors, the answer is vector so, R is a vector very good, what did you tell us in R? magnitude, we are very happy This is the direction of R If it is in the parallelogram and it is diagonal then it will be at 40 degree Will it be at 45 degree?

Sure? It will be at 45 degree, think about it Why? Why will it be at 45 degree?

There is no guarantee of being at 45 degree Now here comes an important point that this R is a vector in itself So, when we have a vector, then there will be two characteristics of resultant. One is magnitude of resultant, which you have learnt to calculate. And the direction of resultant.

You don't know about this by looking at the figure. You don't have much time to make the figure and measure the angle. So, how to calculate the direction of R? Let's learn this also.

Okay? So, as we know there are two vectors, one is vector A and one is vector B. I will prove all the combinations now. And the angle between them is theta.

We have made a valogram. And this is the resultant R. So, we give the direction of R from vector A. From where do we give the direction of R? From vector A.

R is slightly higher than A How high is it? I have said that alpha angle is higher I have understood How high is it? Alpha angle is higher If I know this alpha Then the direction of R will be fixed How high will be A?

Alpha will be higher And the value is known So the direction of R is from vector A to alpha from vector A to angle so the formula will be tan of alpha is equal to B sin theta upon A plus B cos theta ok we will note it and remember it 11th is started Paul said that it will be a great day in life you will have the spring of physics now it has started ok it is fun it is fun and good light weight is good mic will also come in few days it will be good and beautiful ok So tan alpha is equal to b sin theta upon a plus b plus theta. Why do we take tan alpha? Why do we take tan alpha? To take direction. Direction of residue.

We know the magnitude of the result. Let me tell you the formula of magnitude. I will write it again. r square is equal to a square plus b square plus 2ab cos theta.

Very good. And what is the direction obtained from tan alpha? The formula is b sin theta upon a plus b cos theta. Let me explain you one thing. Let's assume that we will not know the direction from a.

We will know it from b. we will not tell the direction with A, we have two vectors we have to tell the answer, we know the answer B value now the direction of the answer is to tell the angle with any of these vectors so you told with A, so this formula is made if you tell with B, so what angle will you use? you will use beta, so tan beta tell yourself what will happen, tell tell you have studied so much math turn it upside down, tan beta What will be the equation of theta?

a sin theta upon b plus a cos theta We don't have to rectify that we can only tell from a No, we can tell from b What will be the formula of tan theta? We will tell from the formula a sin theta upon b plus a cos theta if you keep a formula then your brain will get confused you are a small kid so keep a formula in your mind then you will apply the converse formula then you can tell the angle from B vector or if you want to tell the angle from B then you have to use your brain if you know theta and alpha then you have to find So, how to solve the equation of 10 alpha is equal to b sin theta upon a plus b cos theta? So, how to solve the equation of 10 alpha is equal to b sin theta upon a plus b cos theta? Let us derive the equation of 10 alpha is equal to b sin theta upon a plus b cos theta. both formulas first we will do derivation r square is equal to a square plus b square plus 2ab cos theta because we don't just teach for y,t,j and nil we teach all the basics of 11 so derivation is very important come let's understand how to do it so we will use law in this we will assume that we have two vectors vector a vector b and between them is angle theta and the resultant is r these things we have given make a,b as you want I will make it a little beautiful beautiful all my dear children I made vector A and vector B and vector B I made it as I wanted it can be made here or here it can be made anywhere this is angle theta remember one thing theta join it tail to tail But you will be very clear about the questions that will come next.

What did you say? Take theta angle only when both vectors are connected from tail to tail. Okay, now make imaginary parallelogram.

Imaginary... It is done. This is resulted.

R Now I want R's value. And I have to get this formula. I am doing one thing, I am extending it further. I have mixed it up and name this parallelogram PQ and we have already changed the R let's do this MNOP and what I did is I dropped a perpendicular from O here I have dropped a perpendicular from O point and called it as cube Do you understand?

I have made a parallelogram I have dropped a perpendicular from O point and called it as cube suppose we have a parallelogram, what is a parallelogram? the pair of opposite sides of parallel and equal means if this is B then this will also be B and if this is A then this will also be A means if this side is B then this side will also be A the length of this side will be equal to the length of B vector so we can call this B ok ok, now see from here this angle is theta I will show here if this angle is theta then this angle will also be theta corresponding angle this is theta and this is also theta because both sides are parallel and this is the same line so I can say that this angle will also be theta Very good. Now see a small triangle. Small, beautiful, small, small, P, O, Q triangle. Can you see clearly?

Now I will ask you that what is the value of PQ? How will you tell? We will use our brain. What is the value of cos theta, guys?

cos theta is base upon hypotenuse what is base of this triangle? pq what is hypotenuse? po cos theta is equal to base which is pq upon hypotenuse which is po from here pq will be po into cos theta and how much is po? b so I got b cos theta which means this small length pq PQ which is small in length will be our P cos theta How to find it?

cos theta is equal to base upon hypotenuse Base upon hypotenuse, base is equal to P cos theta Ok, clear? Then I said I also have to find OQ I also have to find OQ So I said in this same triangle, within the same triangle You tell me what will be the value of sin theta? What is sin theta? perpendicular to hypertenes so perpendicular to what?

oq and how much is hypertenes? po perpendicular to? hypertenes ok so I don't know the value of oq and what is the value of po?

b is equal to sin theta so what is the value of oq? b sin theta so what will be this value? b sin theta I can see fear in your eyes To make the poster fall is my goal This is the goal of my life To make the poster fall The children who are scared and distressed I will not let them be scared I will make it simple These two maths can be taught by anyone How to catch it in a short time Suppose you see a vector tomorrow B and this angle is theta listen carefully if I have dropped this perpendicular then the one below will always be B cos theta and this one will be B sin theta we call this as component ok, we will come to this later I have explained it just now if this is B and this angle is theta then B cos theta and B sin theta I will explain it a little more like suppose Listen carefully, this is B and this angle is theta.

Now listen carefully, the angle theta is perpendicular to that angle, and this distance will be B cos theta, and the perpendicular distance will be B sin theta. That means, the angle theta is on the base, B cos theta, and the angle perpendicular is B sin theta. where angle is theta, there is cos theta and in front of it is sin theta where angle is theta, which angle is theta, on this line, how much will it be?

b cos theta and how much will it be on its perpendicular? b sin theta suppose this is r, r is making theta angle from this line, so how much will it be here? r cos theta and how much will it be on its perpendicular line?

r's component, r sin theta, means this This one, R sin theta and base value R cos theta. Remember this. You don't need to tell this in the exam. If you want to say straight, this is V, then say V cos theta. If you want to say straight, this is V sin theta.

We will derive the main formula. You have to pay attention. Now you have to take triangle OQM This triangle is at 90 degree So this is right angle triangle We can apply Pythagorean theorem on this Now we will say MO square is equal to perpendicular OQ square plus base MQ square Can we apply or not?

Now what is the value of MO? are value. Here we are talking about value only, what is the value of the diagonal?

So, what is the value of MO? R square. What is the value of OQ? B sin theta. So, B sin theta whole square.

and how much is MQ? from here to here what is the value? A and this is B cos theta so how much is MQ?

A plus B cos theta squared come so from here it will be B square sin square theta expand it a square plus b square cos square theta a square plus b square plus 2ab cos theta this is coming from this this is coming from b this is the whole square expansion of a plus b a square plus b square plus 2ab and here comes cos theta so do one thing from this term and from this term take b square as common So what will be left? Si square theta plus cos square theta. so 1 will be equal to, here I am doing it, first write a square then b square is taken as common so what is left? sin square theta plus cos square theta and here I have 2ab cos theta now sin square theta plus cos square theta, how much will be my? 1 will be done so a square plus b square and what is this?

2ab cos theta, whose equal is it? I think you understood it very well that we have taken this triangle, the value of R is the square of the aperture which is equal to the square of the perpendicular plus the base is the square of the aperture we have expanded it by a square plus b square and from this we have taken b square commonly and derived the formula. Very good.

Come, let us derive the second formula. in which I had to tell that the resultant has different angles where is the resultant? Direction so, our next formula is Direction of resultant The ocean will fill with drops Slowly slowly you will understand the physics 10 alpha formula b sin theta upon a plus b cos theta Come let's make the same diagram Here a is made Here b is made Imaginary parallelogram is made This angle is theta, from here we have made a resultant r This is called point M, this is called point N, this is called point O, this is called point P From here we will make a perpendicular on this line This is called point Q, this is point B, so this is also point B This angle is theta, so this angle is also theta Now tell me one thing, where b makes theta angle, how much value comes on that side? b cos theta, means its value will be b cos theta and the value of the perpendicular line of this line will be b sin theta Now see the triangle OQM Now this is the resultant And we know the angle from A to A Resultant is a vector and the direction of the vector is given by angle How much angle is it making from the vector A?

How much angle is it making from the vector A? Alpha Now look at this angle OQM Can we get tan alpha value? What is tan alpha in this angle? Perpendicular upon base Look at this angle, who is the perpendicular?

I think it is OQ Who is the base? Thank you. and that will be because it will be of value b sin theta and mq value is a and b cos theta so a plus b cos theta is m so these were two small derivations which are asked in the level and after derivation you feel that you have got it so remember to make a diagonal and extend it so if you have got it then it is good if you have not got it then you should repeat it So kids, we have done the derivation.

Now, first of all, Let's ask some questions. We will ask some good questions and some easy questions. Let's ask some easy questions first.

So, he told us, two forces of magnitude 6 N each act at a point. as shown find the resulting and we have to find the magnitude and direction of the resultant so I will tell you that this is the point and here is a force of 6 Newton and here is a force of 6 Newton and here is an angle of 60 degree now you have to find the resultant of this we have to find the resultant of this and I will give you the option and your option will be 6 root 3 at 30 degree 3 root 3 at 30 degree 2 root 3 at 45 degree Option D we will take 6 at 45 degree Come lets solve this So guys, the result will be from here. If you make a diagram in the beginning, you will enjoy it. The result will be R square is equal to A square plus B square plus 2 AB cos 3. We can write it very easily.

A's value is 6 square plus 6 square plus 2 into 6 into 6 into cos 60. Very good. 36 plus 36 plus, how much will this be? 2 into 6 into 6 and cos 60's value is 1 by 2. So here comes the do. So, I can say 6 square 6 square 3 times.

So, 3 multiplied by 6 square. Let's write 6 square 6 square 6 square. R square value. From here, what will be the value of R?

We will under root it. Under root of 3 into 6 square. So, 6 times square means 6 will come out.

So, what will be left? Root 3. So, what will be the resultant value? 6 root 3. There is only one option where 6 root 3 is there.

But still, for the satisfaction of our heart, we will also take out this angle alpha. So, tan alpha is equal to what? together.

He's 90 da. upon a plus b cos theta now sir which is a from this see a is that from which angle is given means this one although value is same as well so 6 into angle is 60 upon 6 plus 6 cos 60, this time a is also 6 and b is also 6 so sin 60 is root 3 divided by 2 divided by 6 plus 6 cos 60 is 1 by 2 so it is 23006, above 3 is root 3 below 23006 and 3 is 9 so it will be root 3 divided by 3 How to rationalize this? What can I write as 3? Root 3 into root 3 Can you see?

Root 3 into root 3 So how much is it? 1 by root 3 So tan alpha is 1 by root 3 So you know when tan alpha is 1 by root 3 When alpha is 30 degree So the answer matches When does tan alpha happen by root 3? When tan alpha is 1 by root 3 So alpha is 30 degree If tan alpha is root 3, then alpha is 60 degree If tan alpha is 1, then alpha is 45 degree and we have to remember that time will come with time and it is good that it is correct you must have understood the example here I would like to tell you one special thing if both vectors are same in magnitude then the resultant will go from the middle it will bisect If the magnitude of both vectors is same then resultant will bisect the angle See how much angle the resultant came at?

30 degree If you want you can memorize this resultant, it is very easy in exam That there are two vectors, one vector is x and other vector is x How much is the reach of angle? 60 degree then the answer is always X root 3 and angle is 30 degree if two vectors are equal at 60 degree then the answer will be X root 3 and this is very common thing and there is no big thing to know about it so let's take more questions the lack of light in our company I am not going to use generator or else the light will be off I am just reading the question Let's take another question Let's try one more question Two vectors of equal magnitude Let's try this one of equal magnitude are added to give resultant which is of same magnitude as the two vectors. Find the angle between two vectors.

You have to tell me what angle is between those two vectors whose resultant is equal to those two vectors Means I have one vector A, one vector B and this is our resultant R What happened is that the resultant value is equal to A and B and A and B are equal to each other. What is this? Two vectors of equal magnitude R added to give resultant which is of same magnitude resultant magnitude is also same so we are asked that what is the angle between these two vectors find the angle between two vectors their angle is theta so all these are magnitude and equal direction is not possible let's calculate it R square is equal to a square plus e square plus 2ab cos theta So, in the place of r, a, b, consider all as x So, r x square x square plus x square plus 2 into x into x cos theta So, x square x square, how much is this? 2x square, how much is this?

2x square cos theta We don't know theta, we have to find out this only Ok, x square x square, look at the other side So, this will be x square minus 2x square, so minus 2 minus x square is equal to 2x square cos theta. Now, I have to get cos theta out of here. So, cos theta is equal to minus x square divided by 2x square. You will cancel x square x square. And, you will cancel it because x has no value of 0. This will be discussed in the exam.

If x is 0, then never cancel these three things in the paper. In Maths, if x is 0, then you cannot cancel it. Anyways, here x is 0, so we cancel it. So, the value of positive theta is minus 1 by 2. Maybe you have not seen this value in your class yet. When does positive theta come minus 1 by 2?

Slowly slowly you will understand. When theta is 120. When angle is 120, then value of cos theta is minus 1 by 2 For this also we will have a chapter on mathematical cube In which we will tell you how to write angle above 90, above 180 For now you can remember that cos theta is minus 1 by 2 When theta is 120 degree So this was easy, good, just information Keep collecting information in your mind You will get new information everyday So this type of questions will be easy for you I will ask you a good question, get ready It is a big question, you will get the answer from inside Let's go out and make it stand Make it stand Sure? Come on Are you sure you are ready?

Or not? Come on We have two wickers When you are making videos and teaching You should be careful To teach like this You should teach basic things It's not fun You should be very careful Two wickers Take any name Whatever you like Take P and Q P and Q As a sum of As a sum of 18 and their resultant is 12. The resultant is perpendicular to smaller of two vectors. Find the...

value of p and q and angle between them ok, so we have got a good question are you feeling uncomfortable? no problem, we will be with you, no tension see carefully, there are two vectors p and q whose sum is 18, understand this, their sum is 18 and the resultant of y is 12 both these things are different if we add the value of p plus q then it will be 18 if we add vector p and vector q then the value will be 12 I think you are understanding the meaning of these things suppose we add 10 and 8 so 18 can be the answer, value wise, only the value is added but in vector, it is possible to add 10 and 8, so it is possible to get 12 because the answer in vector depends on the angle so these are the two information that they have given us and we have learned that we have to find the resultant there is one more information the resultant is perpendicular to smaller of two vectors the resultant is perpendicular to the smaller vector now let's make an icon so which is the smaller vector in it? we will assume that p is smaller vector I have made p as smaller vector and resultant is perpendicular to p resultant is perpendicular to p now tell me one thing should I make q here? No, we can't make it like this. Because how do we make resultant?

With diagonal. So why we have to make Q here? This is one of my favorite question. So why you make Q here?

Now see, is it correct? Now see the chances of getting right At least this looks like a diagonal or not PSA QS Who told you that it can't be like this? Who told you? First it can be like this This, this, this, this Now see this will be a reserquent Now only this is perfect So, we have to keep this approach in mind.

If you make Q here, then automatically the mind will move. Very good. Very beautiful. And this is the angle between them. Now, we have to use two information.

That P and Q have a connection of 18. The vector sum of p and q, that is the resultant, is bar. So let's apply the resultant formula, I think it will be useful. So what is the resultant formula?

r square is equal to a square plus b square plus 2ab cos theta. We don't even know theta, we don't even know r, we don't even know p, we don't even know q. No, we know r, so what will we get from r?

What will you get from this in life? p square plus q square plus 2pq, we don't know anything. I'm sorry, you got nothing what is the second formula?

angle one you know the angle of this how much is the angle of resultant from P? 90 degree you know the value of alpha how much is alpha? 90 degree from whom? from vector P or you can call it A or B whatever you like now tell me formula of tan alpha B sin alpha sin theta upon a plus b cos theta tan alpha tan 90 tan 90 is equal to b sin theta b is q sin theta divided by a a means p p plus q cos theta tan 90 is equal to 10th class Some will have read it as infinite and some as undefined. The one who has read it...

I am ready to respect It is undefined if you will argue with me Ok, this thing is undefined or infinity Whichever you like You can fight in comment Take out rally Whatever This thing is undefined or infinity When the thing below is zero In your calculator do 1 divided by zero It will be written not defined Or it will be written Math error If there is zero in the denominator, then only the undefined infinity comes. This denominator will be zero. We got one very important information that p plus q cos theta will be zero.

That's it. This is the information we got. P plus Q is equal to 0. So, can I say that Q is equal to minus P.

Q is equal to minus P. I will take this formula. 12 square is equal to P square plus Q square plus 2P.

And I can put Q is equal to minus P. Q is equal to minus b It looks like a simple thing. P square plus Q square minus 2P square. So this will be Q square minus P square.

Will it be possible? Yes, tell me. Q square minus P square is equal to 12 square. Okay, tell me further. What can I write as Q square minus P square?

Look here. I can write Q square minus P square as Q minus P Q plus P. Yes.

Yeah. a square minus b square a minus b, a plus b is equal to 12 square that is 144 now they have given the value of q plus p 18 now it is time to use this q minus p into 18 is equal to 144 18, 8 is 144 so we got the value of q minus p 1 well done and the value of q plus p is 18 Question Time I will add both of them 2Q is equal to 26 So what is the value of Q? 13 Q is 13 so P is 5 13 5 18 Q is 13 So P is 5 So what is the value of 13 and 5?

18 Satisfy sir Great question, great method Very good question Will you take out theta also? Yes, we will take out theta also Why we have kept cos theta Minus what? Now let's use that.

Let's use that. Q cos theta is equal to minus b. Q cos theta is equal to minus b.

How much did Q equalize to? 13. cos theta is equal to minus b. b is equal to 5. cos theta is equal to minus 5 divided by 13. Just leave the answer here.

Now you can't remember all angles of 5 by 13 So you have to remember that There is an angle theta whose cos theta is minus 5 by 13 Will we do this in mathematics? If we don't have an emulator in our hands then how many angles will you remember? cos 21, cos 22, cos 27, cos 28 You can't remember all of them So you don't know if there is an angle whose cos theta is minus 5 by 13 You have asked a very good question Let's go Ok, you want to give a homework? Ok, I will give you a similar question in a similar way. I will bring it.

Ok, I was watching DC Panday and I got a similar question. This question is very old. DC Panday has a similar question. So, a very similar question. Try it once.

See, it is fun. I hope it should be done. Take the name of God. Your own. God will help you.

Stay happy, do meditation in the morning Do something in life I am talking to you, find the question And I got the question We have been told that the sum of two forces at a point is 16 Newton The sum of two forces at a point is 16 Newton This is the sum, not the result if The resultant is normal to smaller force and has value, Resulted t value is 8 Newton find two forces Two forces and option given to them 6,10 8,8 4,12 2,14 So, sum is 16. 6, 10, 18, 16, 4, 12, 16, 2, and? 14, B. Now, try it. See how the question will be.

Try it yourself. Okay? Do it like this. Make a small vector here, A.

And make another vector here, B. And make this resultant R. What will be the angle? 20. And what will be this angle?

Theta. Then you will put the formula of tan alpha. Which angle is alpha?

This one. Resultant angle is with A. Tan alpha, what will you say from this? B sin.

I will help you a little bit 10 alpha is 90 and 10 90 is infinity If this is undefined then the term below will be 0 So a plus b cos theta is equal to 0 b cos theta is equal to minus a R square is equal to a square plus b square plus 2ab we will put the value of r we will put the value of r so we will have r square is equal to a square plus b square plus 2a and b is the cost at this place minus a so it will become b square minus 2 square a square so b square minus a square is equal to 8 so b minus a and whole b plus a is equal to 8 square means 64 so b minus a into b plus a is equal to so now we will solve the first two ok now we will solve it by ourselves try the question yourself a plus b will give you a minus b value a plus b will give you 16 a minus b will give you a minus b value so guys these were the questions on vector addition i hope you liked it and i am thinking that We have time. So I will explain you about subtraction of vector. Time is left. So we will do subtraction of vector today. So I will give you the concept of subtraction of vector.

Wait, I will ask you a question. Find resultant. Let me see how many people do it correctly. And all of you pause and find it. Find resultant of two vectors.

show ok now we discuss now we pause we have one vector 6 one vector 8 Our vector is 8, angle is 120, and I have told you the value of cos 120, so don't worry. Minus 1 by 2, now you have to find the resultant value. Okay, everyone find it.

Pause and find it. I am writing the option here. You will get the option that you have not got it. The question is very important. Very, very important.

Okay? So you will get the answer that you have not got it. I will quickly answer it.

Ok, tell me I hope you all have asked the same question Ok, now try it So what will be the formula? r square is equal to a square plus b square plus 2ab cos theta You remember the formula, so you can tell 6 square plus 8 square plus 2 into 6 into 8 into cos 120 minus 1 by 2 2 from 2 is dead 6 square 36, 8 square 64 Here minus will be done because it is minus minus 6 to 8 is 48, so 100 minus 48 Answer is 52 r square's value is 52, so r's value under root is 52 Very very good. And the answer is completely wrong. What did we tell you? We told you that theta angle is to be joined from tail to tail.

And you ignored this. Here see, the tail of the other vector is joined from the head of this vector. That is head-tail matter.

It is not confused. Tail is joined from tail. Tal is joined from tal.

Tail is joined from tail. Wrong. You have got a wrong question. You have made this question wrong in its entirety. Come, let's make the diagram again.

6. Now I will put this vector here. 8. Everything is allowed. Parallel shifting of vector is allowed.

I can put it from here to here. You can increase this vector and move forward In the first video I taught you how to find the angle between two vectors So watch the previous video of vector Then you will know what is the angle between two vectors And how to find the angle if not given So either increase this or decrease this Now tell me children This angle is 120, this angle is 60, this angle is 60, this angle is 60 Actually, the angle between these two vectors is 60 If you want, you can increase this vector and increase it further This is also allowed, I increased it to parallel Now, what will be this angle? 60 If you didn't get this, then don't cry, watch the previous video, I have told you this with a lot of love and love. Now see, angle is 60, now r square is equal to, yes you are laughing, now anyone will find out sir.

2 into 6 into 8 into cos 60, tell the value of cos 60. Cos 60's value is 1 by 2, 2 multiplied by 2 is 68, 48, this is 100 right. Plus 48, 148, r square, so the value of r and the root is 148. Congratulations if you had hit the C, in the moment you will definitely lie, sir I had hit the C.

Transcript for:Understanding the Basics of Vectors

Transcript for:
Understanding the Basics of Vectors