 So, we are going to begin with a new chapter today that is vector algebra, not a new concept to you. So, in today's class you will find that most of the things you are already aware of and in those spaces I will be a little faster in order to save your and my time, okay? So, I will just begin with a broader level understanding about vectors, most of the physical quantities that we see around us, okay? Most of the physical quantities that we see around us can be broadly classified as scalars and vectors, scalars and vectors. Let me tell you, there is a third quantity as well which is mostly referred to as tensor quantities, okay? Now, scalar and vectors themselves are tensor quantities, normally we say a scalar is a tensor quantity of rank 0, vector is a scalar quantity of rank 1, there is something called dyad which is a tensor of rank 2, there is a triad which is tensor of rank 3 and so on, okay? But I am not going to talk about those because that anyways you are going to study later on in your and I got courses. So, meanwhile how do you define a scalar quantity, any physical quantity which can be represented only by stating its magnitude, okay? So, any magnitude is sufficient enough to state it, example, example, mass, distance, speed, temperature, money, etc., okay? As those quantities which require magnitude, direction, also called as sense, right? Can I say these two parameters are sufficient enough for a quantity to be called a vector quantity? Yes, no? Yes, sir. Okay, what about current? Current has magnitude as well as direction, is it a vector quantity? Is current a vector quantity? Definitely not, right? So these two are not sufficient enough, they are necessary but they are not sufficient enough. You need a third parameter as well which we call as the vectors must follow the parallelogram law of addition which currents don't follow. For example, you must have all heard about KCL, correct? So, let's say there are two currents coming up like this of 5 ampere and 3 ampere and let's say the resultant of these two is going in this wire. So you know that 8 ampere is going to flow in this wire. So it's just a algebraic sum that you're doing, okay? So it doesn't depend upon the direction of these wires. So for example, if I have 5 like this and let's say if I have 3 like this, okay? And there is an outlet over here, this still will carry 8 ampere. So it doesn't depend upon what is the direction of these wires, okay? These currents will still algebraically add up to give you 8, okay? So they don't follow the parallelogram law of addition, hence current cannot be called as vector quantities, okay? So remember, a vector should have magnitude, direction and these two are not sufficient enough it must also follow the parallelogram law of addition or triangular law of addition, whatever you call or polygon law of addition, okay? Examples are displacement, okay? Most of the vectors you would have already come across in your physics, displacement, force, velocity, okay? These are vector quantities. How do we represent a vector? How do we represent a vector? We represent a vector by using a small alphabet with an arrow on top, okay? So A with an arrow on top. Or if you don't put an arrow, mostly in books they don't put an arrow, then they would write it in bold, okay? You could also state a vector by stating its initial and the terminal position. So for example, if a vector starts from A, ends at B and the sense is from A to B, then we call it as AB vector. If the sense is from B to A, let's say if your vector is like this, then you would call it as BA vector. So here the initial point is your B and terminal point is your A. So this is called the initial point and this is called the terminal point, okay? Now the three important things a vector should have, what we call as a characteristic of a vector, okay? So vector has the following three characteristics. One, it has got a magnitude, okay? Which we represent as the vector written within two parallel lines. It actually gives the length of that line which you are using. It is synonymous with the length of the line that you are using to represent the vector. So longer the length of the line you have to make, the longer is the more magnitude that vector has. For example, if I make these two vectors where this length is double of this length, so let's say this length is 1 centimeter and let's say this is 2 centimeter, then this vector that is vector B would be, let's say vector B is parallel to vector A, then vector B will be double of vector A, okay? So it is double the magnitude of vector A. Direction is also the same in this case, okay? So magnitude is nothing but it gives you an idea about the strength of that vector quantity, okay? Second thing that it has is sense or direction which I have already stated, okay? Sense or direction. And the third thing is your support. Now what is support? Support is basically the infinite line from which the vector has been carved out, okay? So it is a line from which the vector has been carved out. The vector is taken. For example, if you say AB vector like this, this AB vector belongs to an infinite line like this, sorry. It belongs to an infinite line like this, correct? So from this infinite line, you have taken out a vector like this which you are calling as your AB vector. Please note that this infinite line will be called as the support. Not very important concept but you should be knowing about the names involved while you are talking about vector quantities. So this is called a support, okay? Next, we are going to talk about types of vectors. So I will be slightly faster because you are only aware of all these concepts. Types of vectors. I will first start with zero vector also called as the null vector. What is a zero vector? Something whose magnitude is zero, okay? But direction is undefined. So a zero vector is a vector whose magnitude is zero. Direction is not defined. It is just like a zero polynomial, right? What is the degree of a zero polynomial? Undefined. What is the argument of a zero complex number? Undefined. In the same way, zero is also a vector quantity but its direction is not defined, okay? Why do we need it? For example, let's say you walked five kilometers towards north, okay? Then you walked from the same point back five kilometers south, okay? What is your total displacement? You will say zero, right? So you started from the same place, you are back to the same place, correct? So zero vector is your displacement here and the direction of this would be magnitude is zero and direction for this would be undefined, okay? Because if you tell somebody I had zero displacement he has no way to figure out what was your direction of motion. You could have gone in any direction and come back to the same position. Next is unit vector. What is unit vector? Unit vector is basically a vector which is first of all represented by a cap, okay? It is always a vector which is in a direction of a particular vector, let's say vector a and its magnitude is unity, okay? So a unit vector in the direction of a vector a is nothing but that vector which has the same direction as the direction of a but has a magnitude of one, okay? So this vector will have the same sense or same direction as that of a, as that of a, okay? How do we find the unit vector? The unit vector is found by dividing the vector by its magnitude, okay? Some of the well-known unit vectors that you would have already come across would be your i cap, j cap and k cap, okay? i cap is a unit vector in the direction of positive x axis, positive x axis. If you want a unit vector in the direction of negative x axis it will be minus i cap, okay? j cap is the unit vector in the direction of positive y axis and k cap is the unit vector in the direction of positive z axis, okay? Third type of vectors that we are going to talk about, co-initial vectors. What are co-initial vectors? Vectors which have the same initial point. For example, these two are co-initial vectors because they have the same initial point over here, okay? Normally, in order to add vectors we need to make some vectors co-initial, okay? We will talk about this when we are talking about the parallelogram law of addition of vectors. Similarly, vectors can be co-terminus also. What are co-terminus vectors? Co-terminus vectors means they have the same terminal point. Let's say a, b, okay? They have the same terminal point, let's say p. Normally, you are talking about co-terminus vectors when you are dealing with the forces acting on a body, okay? So when you draw two forces acting on a mass m, let's say one force is acting like this, other force is acting like this, they become co-terminus. Next, equal vectors. So two vectors a and b are said to be equal, are said to be equal. When number one, their magnitudes are same. Somebody is typing something. Oh, is it so? But I can hear my audio properly. Because I am facing the problem. Even I am facing the problem. Oh, is it because of the bad weather? Because it's raining over here. There's a background sound from your side. Okay, I understood what is the background sound. One second. Is it clear now? Yes, sir. Yes, sir. Okay. Yeah, let me see who all our submitters are also there. Okay. Yeah, when are two vectors said to be equal when their modulus are or when their moduli are same and secondly, their sense must also be the same. Direction must also be the same. Direction is also the same, right? For example, a and b vector of the same length, same direction, then a and b vectors will be said to be equal. So vector a will be equal to vector b. Okay. Next, like and unlike vectors. Okay. What's the like and unlike vectors? Two parallel vectors which have the same direction are called like vectors. So we say a and b are like vectors. Are like vectors when they have the same direction. And they have the same direction. Okay. Their magnitudes need not be equal. For example, a can be like this, b could be like this. Okay. So where we say these two are like vectors. Okay. Similarly, a and b would be called as unlike vectors. Okay. When these, no, remember that when they have same direction, I mean to say they are parallel and that sense is same. Unlike vectors, a and b would be parallel, but their direction would be reverse. We have reverse direction. Getting the point. So let's say is like this, b is like this. Then in this case, a and b would be unlike vectors. Okay. Next. Next is this is very important. The concept of free vectors and localized vectors, free vectors and localized vectors. What is a free vector? Free vector is something which has no fixed initial point and terminal point. For example, a vector a like this, if it is free, then if you draw it like this also, it will be the same. If you draw it like this also, it will be the same. It will all be a vector. Okay. So this is free to move. Okay. Now, for example, displacement. Okay. If I say I'm going five kilometers towards north. Correct. It doesn't matter whether I'm starting my journey from Bihar or I'm starting my journey from Punjab or I'm starting my journey from Bangalore. All these vectors, even if I go five kilometers north would be considered to be the same. It doesn't actually matter from where I'm starting my journey. It has no fixed. So free vectors, they don't have fixed initial point. So no specific initial point. And since there's no specific initial point, there's no specific terminal point also. Okay. But remember the magnitude and the sense must remain the same throughout. Okay. That means you can translate, you can take them parallel to themselves anywhere in space, but you can't change their direction and you can't change their magnitude. Locally, by the way, let me tell you in maths you always deal with three vectors. Okay. In maths, your vectors are free. Okay. So use in mathematics. In maths, we normally take a vector parallel to itself anywhere we want. Whereas localized vectors, they have fixed or specific initial points. An example of it would be a simple example of weight of a body. Okay. So let's say there's a body like this. Let's say it's weight force. It will always start from the center of mass of the body. Correct. It will always start from the center of mass. It cannot go anywhere. For example, I cannot draw a body over here and say this is the weight force of the body. Correct. It will belong because this is a localized force. It will always start from the body. If you attach a string to the body also. Okay. Its direction will always be opposite starting from the object and going the other way around. Isn't it? So tension is a localized force. Okay. So localized four vectors are mostly use in physics. Use in physics. Okay. So unless central stated, in fact, in all the cases that we are going to deal with, we are going to talk about free vectors, not localized vectors. Any questions so far? No sir. Okay.