 Now I'm going to quickly jump into the actual operations on vectors, operations on vectors, fine. So operation on vectors we'll first talk about addition, addition of vectors and I think you have done this a couple of times to find resultant that's why we call it as finding the resultant of vectors also, okay? Addition is as good as finding the resultant of vectors. So when you add two vectors A and B and you get a vector C from it, we call C as the resultant of A and B. Now how do we add two vectors? There are multiple ways to do it. The first way that we are going to talk about is the parallelogram law of addition. So first method is a parallelogram law of addition. Suitable for those cases where your vector is localized, okay? But it can also be applied to free vectors, okay? So parallelogram law of addition is used when your vectors are localized. For example, if you want to, if there's a, if there are multiple forces acting on a body, that's it is a body of mass M and force F1 is acting in this way, force F2 is acting in this way and you want to find out the resultant of these two forces, okay? Then we can use the parallelogram law of addition. This law says that if you have two vectors, let's say vector A is like this and vector B is like this. Let me draw it slightly. Yeah. If you want to add them or if you want to find the resultant of these two vectors, first make them co-initial, okay? How do you make them co-initial? By connecting their initial points. So this is the initial point of A. This is the initial point of B. Let me make it co-initial. So if you make it co-initial, you have to either take A parallel to itself so that it comes over here or take B parallel to itself so that it comes over here, okay? Any of the two you can do. So let's say I take B vector parallel to itself. So A is like this. B, I brought it like this, okay? Remember I have not changed their magnitude and direction. So this law says that complete a parallelogram, complete a parallelogram such that these two vectors form the adjacent sites of the parallelogram. Once you have completed the parallelogram, a vector originating from the common co-initial point going till this opposite end, that vector would be called your resultant of A and B, okay? Couple of things here to be noted that this A plus B will always be having a magnitude less than equal to the sum of the individual magnitudes of A and B, okay? This is because of the triangle being formed over here. So as you can see, this is your vector A, okay? So if you see this triangle, A plus B is magnitude of A plus B is the length of the third site which will always be less than the sum of the other two sites, okay? This is also called the triangle inequality, okay? Next method that we normally adopt to add vectors is called the triangle law or triangular law of addition, okay? In this, what do we do is we connect the terminal point of one with the initial point of the other. So let me take the same two cases. This is your vector A and this is your vector B, okay? So what do we do? We connect the terminal point of A to the initial point of B. You can do the other way around also. It doesn't make a difference. You can connect the terminal point of B with the initial point of A. So let me do that. So let's say A goes like this, B is like this, okay? B is like this. Then a vector starting from the initial point of A and going to the terminal point of B, that vector would be your A plus B vector. That means you're completing a triangle by doing this, okay? That's why it is called the triangular law of addition of vectors, okay? And same inequality holds true in this case as well. Now triangular law of addition is actually a very specific case of what we call as the polygon law of addition. Polygon law of addition is suitable for those cases where you have more than two vectors involved, okay? More than two vectors involved, okay? For example, if I have a vector like this A, any other vector B, and other vector like this C, okay? If I want to add these vectors, what I do is, again, I start connecting the terminal of one with the initial of the other, terminal of one with the initial of the other. And this you can do it in any order. It doesn't have to start from A. You can start with B also. You can start with C also. But just keep connecting the terminal of one with the initial of the other. For that, you need to move these vectors around also. So let's say I connect A, then I connect B, then I connect C. Let's say C, okay? So triangular polygon law says that the resultant would be a vector connecting the initial point of the first vector to the terminal point of the last vector. And normally in your books and all, they would draw this with dotted lines, okay? This line represents that this is the resultant of all these, okay? So this vector will be your A plus B plus C, okay? Couple of properties that we need to know about vector addition, properties of vector addition. First property is vector addition is commutative. It doesn't matter whether you're doing A plus B or B plus A. They are commutative. Secondly, they are associative. So it doesn't matter whether you add A plus B and then add the resultant to C or whether you add A to the resultant of B and C, okay? So this is associative. Third, zero vector is called the identity vector, okay? So zero is called here. Zero is the additive identity. These terms have actually evolved because vectors form a very important vertical under linear algebra, okay? So you need to know this however there's no direct uses of it. Then there's something called additive inverse. So my negative of a vector is called its additive inverse. So we say my negative A is the additive inverse, additive inverse of vector A, okay? And I've already told you the triangle inequality that is A plus B mod will always be less than equal to mod A plus mod B. Equality will hold true when your A and B are in the same direction, okay? Equality will hold when your A and B will be in the same direction. Okay? Now when we talk about subtraction of vectors, we don't have to deal separately with subtraction of vectors because subtraction is actually covered under addition of A with negative of B. Now what is negative of B? Negative of any vector is nothing but it's a vector with the same magnitude. Let's say negative of a vector, let's say I call it the vector as vector P, okay? Negative of a vector P is a vector which is having the same magnitude as P but direction is reversed. But the direction of, but the direction is reverse of P, okay? I'm going to talk more about it in the scalar multiplication. The direction is reverse of P, okay? For example, if I say vector A is going 10 kilometers towards north, then negative A is nothing but going 10 kilometers towards south, okay? Simple as that. So when you are subtracting two vectors, you are adding A with the negative of B, okay? So let's say these are my vectors A, okay? And this is my vector B. I want to find A minus B. I want to find A minus B. So what do I do is first of all, I reverse the direction of this vector. So that means let me take the reverse of vector B which is like this. Then I can follow any of the two laws that we have seen whether triangle law of addition or parallelogram law of addition doesn't matter. So let's say I follow the parallelogram law of addition. So first I have to make them co-initial, okay? So this is the initial point of A. This is the initial point of B minus B. So I have to make them co-initial. So let me do it like this, okay? This is my A and this is my minus B. Then I complete a parallelogram like this, okay? This vector here would be your A minus B vector, okay? Now let me show you something very interesting. Had you completed a parallelogram with A and B itself, let's say I complete a parallelogram with A and B itself, okay? Of course this diagonal will represent A plus B, okay? This is your A plus B vector, okay? And you would be happy to know that in the same parallelogram, had you connected the other diagonal, as you can see this length and this length are exactly the same. Now the direction has to be from the B point towards the A vector here. This vector will now represent a A minus B vector. So in the same parallelogram, one of the diagonal represents A plus B, other diagonal represents A minus B, okay? Now I'm going to begin with something which is very important for problem solving position vectors, okay? What are position vectors? First of all, before I talk about them, let me tell you these are the concepts which are going to connect vectors, which are going to relate vectors to coordinate geometry. That's why it becomes very important that we understand this concept of position vector very, very carefully because many of the geometrical theorems or proofs can be done through vectors if we are able to understand what is position vector, okay? Now what is position vector? Position vector is a vector which connects the position of a point with a reference position, okay? With a reference position, which we normally call as the zero vector also or the origin. So what is origin in case of coordinate geometry is the reference point or reference vector in case of vectors. So a vector which connects O to P like this, this vector would be called as the position vector. This vector would be called as the position vector of P, position vector of P, okay? And normally we write it like this. Let's say I call it as vector P. So this is your P vector, okay? Now let me tell you this is a very, very, I would say rare way of representing a point's position vector. Normally they just say P having a position vector of this. They will never show you the origin, okay? Origin is up to you to choose. See it's just like mentioning a point. Let's say I say there's a point 1, 2. Do I tell you that where is the origin? No, right? You choose your origin and accordingly you make a point 1, 2. So origin position is never specified. It is up to you to choose your origin position, correct? Wherever you choose the origin position, relative to that you choose 1, 2, correct? In the same way if I say the position vector of a point is let's say P vector, I would never say where is my origin. It is up to you to choose what is the position of the origin, okay? I will just specify something like this. P has a position vector of P, okay? So I will say something like this. P has a position vector of P, Q has a position vector of Q, okay? Now let me ask this as a question. What will be P Q vector then? Can somebody tell me what is P Q vector in terms of small p and small q? Sir, are you asking the distance between p and q? No, no, I'm talking about the vector p and q. Come on, I think you've done this chapter in school, right? Okay, I've got some response. Q minus p, Shia says Q minus p. Why not P minus q? How do you know it's Q minus p? Sir, they have the same initial point origin so we need to take minus p in order to do it. Okay, see let's say I take the origin over here, okay? Now it's up to me where I take this origin, okay? So when I say position vector of P is P, what does it mean? It means that a vector connecting, a vector connecting O to P is this P vector, correct? A vector connecting O to Q is your Q vector, correct? Yeah. Now if you want to think as if you want to travel from P to Q, okay? And the only roads available are this road and this road. See the motion of my cursor. You can only take the known roads, this road and this road, correct? So if you want to go from or you can use your vector addition, see vector addition is nothing but trying to figure out alternative way to accomplish that vector path, okay? So what I normally tell people who have not been introduced to vector addition is that let's say you're trying to go from P to Q. So I'll take the path from P to O first, okay? Now when you're going from P to O you are traveling minus P vector, correct? Because you're going in the reverse direction of P. Now you're going from O to Q. O to Q means you're going in the direction of Q. So basically this will become Q minus P as you rightly said, okay? Other way of looking at it is let's say I consider that O P plus P Q will give you Q. Look at the way the arrows have been framed. O P plus P Q will end up giving you O Q, okay? So P plus let's say P Q is a known vector. This will be equal to Q. So from here also we end up getting the same answer. Now in order to facilitate remembering of this normally I say if you want to go from A to B it's nothing but position vector of B minus position vector of A. Just remember this formula, okay? I call this as destination minus source. So wherever you're going that is your destination. From where you're coming that is your source. So A B is nothing but destination minus the source position vector. Okay? Now if this concept is clear we'll take few problems on this but before that I would like you to solve a simple question. Let's say this is a regular hexagon A B C D E F, okay? Let's say A B vector is A, B C vector is B, okay? Find the following vectors in terms of A and B. One is C D vector. So tell me what is C D vector, D vector, E F vector and F A vector. In terms of A and B express these following vectors. Remember it's a regular hexagon. I'm sure at least D vector and E F vector you can say very easily, correct? Yes. What is that? D will be minus A. Correct. D it is minus A. E F will be? Minus B. C D if you can find out the negative of C D will be F A, right? But how do you find C D? Sir can we say that the sum of this whole thing will be like zero? So it doesn't help you because C D and minus F A will cancel, A B and D will be cancelling, B C and E F will be cancelling. So how does it end up giving you the value of C D and E F? Yes. Any idea? Okay let me help you out. Would you all agree with me that this vector here would be 2B vector because this is in the same direction of B and double of the vector B because of geometry. Okay should I show you how it is double the length? I'm sure you can figure it out. This is 60 degree correct. So this will be let's say this length, whatever is this length, X. This will be X by 2. Similarly here also it will be X by 2. Okay and this is X itself. So X, X by 2, X by 2 will make it to X length. Okay so since the direction is the same and the magnitude is double of that of B, I can call this vector as 2B. Correct? Now let us use the polygon law of addition. So can I say A B plus B C plus C D will be equal to A D. A B plus B C plus C D is equal to A D. A B is A. B C is B. C D not known to me but A D is 2B. So can I say C D vector will be nothing but take it on the other side will become B minus A. So C D is B minus A. So F A would automatically be A minus B. Clear?