Vectors Physics Class 11th

Transcript

[Music] don't worry because in this video I'm going to make the topic of vectors very easy for you we will look at the concept of vectors Vector notation and different types of vectors vectors is all about Direction and I promis to take you in the right direction now before we begin I would like to let you know that we have courses on science maths coding and artificial intelligence on our website and Android app so do check it out the links are given below all right let's get started and dive into the world of vectors first let's talk about scalars and vectors and see the difference between these two scalar quantities are those physical quantities which only have a magnitude they only have a value but no dire ction example of scalers are mass distance time temperature etc for example we can say Mass equals 5 kg or distance equals 3 km or time is 2 hours or the temperature is 20° C these are all scalar quantities they only have a magnitude that is specified by the number here and the units but they don't have any dire ction when you say Mass equal to 5 kg 5 kg is the magnitude the value we don't say 5 kg North or 5 kg South that doesn't make any sense because it is a scalar quantity it has no Direction only a magnitude Vector quantities are those physical quantities that have both magnitude and Direction examples of vectors are displacement velocity Force momentum and so on for example we say that the displacement is 2 km towards the North or velocity is 5 m/s Southeast or force is 10 Newtons towards the east vectors have a magnitude they have a value and a direction sometimes when we say that the force is 10 Newton we are specifying the magnitude but we are ignoring the direction here but this does not mean that force does not have a direction force is a vector quantity it has a direction we just haven't specified it so remember Vector quantities both have a magnitude and a Direction let me ask you a question what is 3 Kg + 4 kg you'll say it's so simple it's 7 kg now let me ask you what is 3 Newton + 4 Newton is it 7 Newton this is true only in the special case where the two forces are in the same direction if the two forces are in different directions we can't simply do 3 Newton + 4 Newton equal to 7 Newton we need to use vector addition scalers can be added according to the laws of algebra but vectors need to be added using the special laws of vector addition so remember scale and vector addition is not the same because these Vector guys are different they have a direction since vectors are different from scalars we need a special Vector notation to represent them for example force is a vector quantity if we want to represent the force Vector we write it as f with an arrow over it or F is written in bold to represent that it's a vector for example we can write f = 10 Newton East this represents the force Vector of magnitude 10 Newton and the force is in the East Direction now if you want to represent only the magnitude of the force Vector then we can write f = 10 nton note now that F does not have an arrow over it or it is not written in bold so it's representing only its magnitude another way to represent the magnitude is by drawing two vertical lines around F so you can write it like this F = 10 NT this is also called the modulus of the vector this is again the magnitude of the force Vector without its direction so it's only the value of the vector vectors can also be represented visually by drawing a straight line with an arrow head at one end the length of the line gives you the magnitude of the vector and the Arrow Head gives you the direction for example this Vector drawn here represents a force of 10 Newton towards the east and this Vector represents a force of 20 Newton towards the north note that it is double in length compared to the 10 Newton Force vector and it points towards the North and this Vector represents a force of 5 new towards the Southwest note that it is half in length compared to the 10 Newton vector and it's pointing towards the Southwest so just like in coordinate geometry we use a scale for the length of the vector for example 1 cm here is 2 Newton so 10 nton Vector can be represented by a line of length 5 cm for 20 Newton we will use a line of 10 cm and so on so the length of the line is indicating how large the vector is the starting point of the vector represented by o here is called the tail of the vector and the ending point represented by a here is called the head of the vector so the vector has a head and a tail and the vector here is o a and it is written with an arrow on top of it remember the arrow represents that we are talking about a vector quantity or sometimes you write OA in bold these are both Vector notations right so either you use the arrow or represent it in bold to indicate that we are talking about a vector quantity now you know the basic concept of vectors let's discuss some important types of vectors such as position and displacement vectors and polar and axial vectors A position Vector is used to describe the location or position of a point or object in space relative to a chosen reference point or origin for example let's say an object a is located at 3A 4 here on the coordinate plane then the vector drawn from the origin o that is 0a 0 to a3a 4 is called the position Vector of the object position Vector has both magnitude and direction now what is the magnitude of the position Vector OA here can you calculate it that's right you can use Pythagoras Theorem and calculate the magnitude or the length of the vector since we have a right angle triangle here so the length of the vector is going to be the hypotenuse which is going to be squ < TK of 3 S + 4 S which equals 5 units so OA has a magnitude of 5 units and the direction of this vector is roughly towards the Northeast Direction you can see here now the position Vector is often denoted by R with an arrow on top of it or R written in bold and its job is to give the position of the object now mathematically this position Vector can be represented as R = 3 I + 4 J where I and J are the unit vectors along the X and Y Direction next let's look at the displacement Vector a displacement Vector is often denoted as delt r or D represents the change in position of an object as it moves from one point to another in space like position vectors displacement vectors also have both magnitude and Direction indicating how far and in which direction an object has moved displacement vectors are not concerned with with the path taken by an object they only describe the change in position from the initial to the final position because remember that is the definition of displacement right the shortest distance from initial to the final point now mathematically the displacement Vector can be found by subtracting the initial position Vector from the final position Vector so we can write delt R = to R final minus r our initial for example if the object was initially at 3A 4 and let's say it moved to b2a 7 then the displacement Vector AB is going to be 2 I + 7 J minus 3 I + 4 J so you subtract the initial position Vector from the final position vector and we get the answer - I + 3 J so that is our displac ment Vector AB here vectors can be divided into two broad categories polar vectors and axial vectors polar vectors are those vectors that have a starting point or a point of application for example displacement and velocity these are polar vectors since they have a starting point force is also a polar Vector since it has a point of application actual vectors are those vectors which represent a rotational effect and act along the axis of rotation based on the right hand thumb rule examples of actual vectors are angular velocity torque and angular momentum the right hand thumb rule for actual vectors reminds me that do hit the like button and share this video with your friends now we will discuss some more important terms that are used in vector algebra and I know there are lot of terms for vectors but don't worry these are simple and I will explain it with some easy examples for you so let's start we'll start with equal vectors two vectors are said to be equal if they have the same magnitude and the same direction for example if two Force vectors are both 10 Newton each and let's say both of them are acting towards the east Direction then the two vector are equal so we can write Vector a is equal to Vector B now what is negative of a vector negative of a vector is another Vector having the same magnitude but the opposite direction so for example if there's a force a equal to 10 nton towards the east and force B is 10 Newton towards the West so if we take the direction towards the east as positive we can write a = 10 nton then B is going to be written as- 10 nton because it's in the opposite direction so according to Vector notation we can write B is equal to minus of a so B is the negative Vector of a here we have discussed modulus of a vector earlier remember modulus of a vector is the magnitude or the length of the vector we represent presented by drawing vertical lines around the vector or writing a without Vector notation magnitude or modulus is a scalar quantity so for the force a equal to 10 nton towards the east the magnitude a is going to be equal to 10 Newton that is just the value without any direction that is basically the modulus of the vector only its magnitude unit vector VOR is a vector of unit magnitude or we can say magnitude of one and it is drawn in the direction of the given Vector a unit Vector can be obtained by dividing any Vector with its magnitude unit Vector is written with a carat symbol on top of it as shown here unit Vector is called a karat a hat or a cap now rearranging the equation we can write any vector as its magnitude times the unit Vector in the same direction so the vector a is going to be the magnitude of a times the unit Vector a hat note that the magnitude of the unit Vector is Unity that is one but the unit Vector has no units or Dimensions fixed Vector is a vector whose initial point is fixed it is also called a localized Vector for for example the position Vector of a particle is a fixed Vector since its initial point is fixed at the origin free Vector is the opposite of a fixed Vector free Vector is a vector whose initial point is not fixed it is also called a non-localized vector for example Force Vector is a free Vector it does not have any fixed initial point because the force can be applied at any point right so the force Vector can move around so that's why it's called a free Vector colinear vectors are those vectors that act along the same line or along parallel lines I'm going to use these two pencils to represent vectors so when will they be collinear vectors if they are acting along the same line like this or along parallel lines so see these two are parallel to each other so they are Co colinear vectors the two colinear vectors that have the same direction are also called parallel vectors now what is the angle between the parallel vectors can you guys tell me so if you look at these two parallel vectors can you see the angle is 0° which will be more obvious if you bring them together like this so if they are along the same line the angle is going to be 0° two colinear vectors having opposite direction so when they are placed like this are called unlike or anti- parallel vectors and what is the angle between the anti parallel vectors here that's right it's going to be a 180° here so these are anti- parallel vectors c-plan of vectors are vectors that act along the same plane so if you have two vectors like this they are lying along this same plane or if the two vectors are like this they're lying along the same plane or same same surface so let's say we have the two vectors like these these will be called co-planar vectors they are lying in the XY plane so you can imagine a XY plane here but if you have a third Vector which is not lying in the same plane a vector like this then this Vector is not co-planar with the other vectors so co-planar vectors means it will also have to be like this and these are see lying on the same plane here Co initial vectors are those vectors that have the same initial point so if I take these two vectors when will they be Co initial when they have the same initial Point like this the starting point of these vectors is the same and remember the starting point is called the tail of the vector so as you can see the two tails are touching each other at the same point so they are called co- initial vectors co-terminus vectors are vectors having a common terminal point so if you have two vectors if they have the same terminal point that is the ending point is same these vectors are called co-terminus vectors and remember the ending point of the vector is called head so we can say the two heads of the two vectors are meeting here and these are co-terminus vectors now let's talk about a very special Vector which is called the zero Vector you might be thinking why are we talking about is zero Vector just like the number zero is very important in mathematics similarly zero Vector is a very important Vector the zero Vector is also known as a null Vector the zero Vector is typically denoted as zero with an arrow on top of it now zero Vector is a vector whose magnitude or length is zero a zero Vector does not have a specific direction or we say it has an arbitrary Direction because it's a zero Vector so where will it point to it is often shown as a point or a dot in the vector diagrams because it doesn't Point anywhere now why do we need a zero Vector let's say you have two equal vectors A and B so can you tell me what is a minus B it will be zero right but when you subtract two vectors you should also get a vector that is why a minus B is going to be equal to the zero Vector what are some other examples where the zero Vector can be used let's say a particle lies on the origin then its position Vector is a zero Vector similarly what will be the velocity Vector of a stationary object that is an object at rest that's right once again it's going to be a zero Vector because it is at rest so zero Vector is important and it has practical application ations let's discuss some mathematical properties of the zero Vector when you add the zero Vector to any other Vector the result is the original Vector itself so for any Vector a a + the zero Vector equals a similarly a minus the zero Vector is also equal to a so zero Vector is the additive identity for vectors now when you multiply a real number by a zero Vector it results in the zero Vector so if K is the real number K * the zero Vector equals the zero Vector because you're multiplying it with the zero Vector now if you take any Vector let's say a and you multiply it by the number Z what do you get that's right you'll get the zero Vector so 0 * the vector a equals the zero Vector if Alpha and beta are two different nonzero real real numbers and Alpha * the vector a equals beta * the vector B then this relation can hold only if a and b are both zero vectors because Alpha and beta are different numbers so the equality will hold only if a and b are both zero vectors the last thing we will look at is the multiplication of a vector by a real number for example if you multiply a vector a with two this results in a new Vector 2 a that is double in magnitude but it has the same direction so if a was a force Vector representing a force of 10 Newton towards the east then 2A is a force Vector representing a force of 20 Newton that is double the magnitude but in the same direction that is towards the east but what will happen if we multiply the vector a with Min - 2 will it give us the same answer no it'll give us a force of magnitude 20 Newton but in the opposite direction because there's a minus sign that is the force will be towards the West so multiplying a vector by a real number is known as scalar multiplication as we are multiplying the vector with a number which is a scalar it results in a new Vector that is related to the original vector by scaling its magnitude and the direction can be same if it's a positive number and the direction will become opposite if it is a negative number so when you multiply a vector a by a real number that is a scalar K the result is a new Vector often denoted as K * the vector a so we write it as k a friends I hope the concept of vectors the notation of vectors the different types of vectors and all the terms that we discussed related to vectors is crystal clear to you now we will look at addition and subtraction of vectors in our next video now make your finger a vector and move it in the direction of the like And subscribe button so do hit the like button and share this video with your friends and don't forget to subscribe to our YouTube channel and click the notification Bell and select the all options so that you don't miss out on 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