Vectors applications | NEET | JEE MAINS |

Ajouté : 2026-05-13

[00:00:00]Okay. So here what are the applications of vectors? We shall discuss about that.

[00:00:05]So you got a basic knowledge about what is the meaning of vector addition vectors and what are the directions, what are the magnitudes, everything you know. But now we shall discuss about what is the meaning of horizontal forcing. Okay. So to understand the horizontal force, let us imagine that this is a pendulum hanging. This is going to be the pendulum hanging downwards. And what did I do? Know I applied a horizontal force in this direction. Okay, this is going to be the horizontal force which is F here. And this pendulum when it is pulled towards horizontal, it will raise to certain angle theta like this. Okay. So, it will raise to certain angle theta like this.

[00:00:42]Then if it is raising to certain angle theta, what is going to act downwards?

[00:00:46]MG is acting downwards. Then what is the resultant of these two? What is going to be the resultant of these two? For example, if I close this and this now like this. If I close these two, okay, and this is going to be the resultant of those two. Okay, this is going to be the resultant. And let us write this as a resultant force. So what is this resultant force? See this is angle theta means this is also going to be angle theta because these are parallel. These can be taken as corresponding angles.

[00:01:15]Okay. So these are corresponding angles.

[00:01:17]So horizontal force is F. This is MG.

[00:01:21]Okay. So if this is F, you know this is also going to be F because opposite sides in a rectangle are going to be same. Okay. So now what I have to do I have to find out what is the horizontal force. Now for that what I'm going to do I can write now tan theta is going to be opposite side that is F and adjacent side is going to be mg. So F is equal to mg tan theta is the horizontal force.

[00:01:49]Sometimes in the problem in the previous year problems I have seen that angle is given as 45° and mass of the pendulum is given as 10 kg. Okay. What is the mass of the pendulum? This is having a mass of 10 kg. And he's asking us to find out what is the horizontal force applied when you raise the pendulum to certain angle is equal to 45°.

[00:02:10]Okay. So what is our aim? You know this is a simple pendulum. You are pushing with sudden you are pulling the pedalum with horizontal force. Then you know mg is always acting downwards. Okay. Then F is in this direction. MG is in this direction. This is going to be the resultant. Okay. So as this is a rectangle, this force and this force are same. This is mg and that is also mg. In a rectangle opposite vectors are going to be same. Okay. So our aim is to find out what is the what is that horizontal force. So tan theta is nothing but opposite side by adjent side that is f by mg. F is equal to mg tan theta. Okay.

[00:02:47]And sometimes he gives you angle as well as mass. And he ask you what is the horizontal force. Very simple that the horizontal force can be taken as mass is how much? 10 into 10 and tan 45 is going to be 1. Therefore horizontal force applied is going to be 100 ntons. Okay.

[00:03:03]This is a simple concept. The next thing is sometimes he'll ask you to find out the tension also. Okay. So how are you going to find out the tension for that?

[00:03:11]Let us imagine this is a pendulum. Okay.

[00:03:14]And you are pulling it with horizontal force F here. And this is going to be mg. If such situation comes then what is going to act here? Now let us see here this is the length of a pendulum is L.

[00:03:27]This is the length of a pendulum which is hanged from a rigid support here.

[00:03:31]Okay. So what is the tension on the tension in the string? Tension is considered as T. This is going to be the tension in this direction. Tension in the string means what? The stiffness created in the string because of weights and forces. Stiffness okay is called as tension there. So now as this is an angle theta here in the upward direction also if you consider this is angle theta then this is going to be P cos theta.

[00:03:57]Okay. And this is horizontal force in this direction it is going to be P sin theta. Okay. Now what are you going to understand from this? As the pendulum is equilibrium condition mean balanced condition then all the forces are going to be balanced here. So what do you understand from this? You will understand that t sin theta in this direction and h in the opposite direction is going to be same. So you can find out the tension if you know the horizontal force by using the formula t is equal to f by sin theta. If you know the force if you don't know the force but you know what is the weight of a body then what you do? you have to apply P cos theta. Okay. So what is P cos theta is in the upward direction and mg is in the downward direction. See these two forces are same equal and opposite action reaction forces. These two forces are going to be same. Weight is an action t cos theta is a reaction.

[00:04:53]Equilibrium position means all the forces acting in the opposite directions will be same. So what are we going to understand from this? This is going to be here. If you want to find out tension and if you know the weight of a body then you have to apply tension. T is going to be mg by cos theta. If you know the force horizontal force we can apply this and get the answer. So you can find out tension if you know the force and this one. Okay. So if this is okay this is the second concept. So what is this?

[00:05:20]This is going to be the second concept.

[00:05:22]Find out the tension and horizontal force. This is going to be the first concept. Then what is the next remaining things regarding the same concept here say here if this is a pendulum in different angle you have to think here you are applying a horizontal force F_sub okay horizontal force F and mg is acting downwards and this is going to be the resultant force if this is length here if I draw a line like this is making 90° this is going to be angle theta this is also going to be angle theta here if this is L here you should know that the component component of that is going to be L cos theta. Am I right or not? The component of that is going to be here L cos theta.

[00:06:01]Understood? L cos theta. Okay. And what is this distance? This is going to be X horizontal displacement taken by a body.

[00:06:10]So what is the horizontal displacement taken by a horizontal force? Because of horizontal force, how much distance horizontal it has been moved which is taken as X here. Okay. And what is in this direction? You have to apply there is a resultant force acting. What is the resultant force here? If this is F and this is mg, resultant force is F² + MG square. Isn't it understood? Then okay.

[00:06:36]So resultant force concordant formula.

[00:06:38]This is the formula to find out the resultant force. Okay. So if this is the situation now there is one identity which you have to remember. What is that identity? Force by magnitude of the force is going to be constant. Okay.

[00:06:52]Force divided by the magnitude applied in a constant. Understand force and the magnitude is always constant. What is the meaning of that? Okay. So you should remember that here. What is the force acting downwards? Mg and what is the magnitude acting in this direction? Can you tell me what is the magnitude? E cos theta I can write now. This is this is square root of L².

[00:07:22]H >> yes this minus L square - X² hypotenus L square okay so roo<unk> of L square - S and the magnitude is mg and sorry force is mg and it magnitude is going to be L² - X² okay and horizontal force direction of force is F and the magnitude is going to be X understand going to be X then here in this direction resultant force is acting what is the resultant force root of F² plus mg whole square and magnitude in that direction is going to be the length of a pendalum. Okay. So from this identity you can find out unknown things. Okay. If you know L and X L X okay L X F MG this is one concept based on this concept one more thing I'm going to explain now which is very important here which is called as lamore. Okay, Lamb's theorem. What is the meaning of Lami's theorem? Here you just see in this direction for example force is f_sub_1.

[00:08:27]In this direction force is f_sub_2. In this direction force is f_sub_3. Okay.

[00:08:33]Opposite to f_sub_1 angle is theta1. Let us imagine opposite to f_sub_2 angle is theta_2. Let us imagine opposite to f_sub_3 angle is theta_3. So what is the lamis theorem explaining here? Lami's theorem. And what do you mean by this lamis theorem which is very important to solve the type of problems. So lamis theorem says that force f_sub_1 divided by s of angle in opposite direction then opposite direction. Next force f_sub_2 this is going to be the force f_sub_2 and opposite to that is angle sin theta_2 and the force is f_sub_3 and opposite that is s of angle of opposite that is theta. Okay. This is going to be lame theorem. By using this lamage theorem, you can solve that problem. How can you solve that problem here? So to understand that problem here, you just simply do one thing. What will you do?

[00:09:26]If you draw one line like this, okay, it is 60 means this is going to be how much? Okay. If this is 60, this is going to be 30. If this is 30, this is going to be 60 because this is 90°, isn't it?

[00:09:38]Okay. That means the total angle, this total angle is going to be 90°. Okay. So I am applying this lamis theorem. Okay.

[00:09:46]Now just see I will apply the lamis theorem here and I will write that 900 is the force which is acting downwards and exactly opposite 90° si 90. Okay.

[00:09:58]And then one more thing is tension t1 is acting and exactly this is going to be 90. This is also going to be 90. So 90 + 60. T1 opposite to that angle is how much? 150. So t_sub_1 / sin 150 is equal to ts2 is in the opposite direction.

[00:10:16]This total angle is going to be 120, isn't it? It's going to be sin 120.

[00:10:20]Okay? Sin 120. And what he's asking us to find out tension t1 by t2 is asking.

[00:10:26]And you know that first of all what will you do? First of all, what will you do?

[00:10:30]You compare these two now. Okay? That is t1 sin 150 is what? Sin 180 - 30 which is sin 30 mean 1x2 isn't it? So t1 / 1x2 is equal to sin 90 is 1 means you can write 900 / 1. So t1 is equal to 900 into half your answer is 450 newton. So I got t1 as 450 nton by using lambis theorem. The next thing what I'm going to find out is t2. So t2 divided by sin 120 means sin 180 - 60 which is nothing but <unk>3 by 2 which is equal to 300 or 900 sorry. So 900 divided by sin 90 which is going to be 1. Therefore ts2 is going to be 900 into <unk>3x 2 which is 21 * 2 450 * so t2 I got it as 4 450 roo<unk>3. Therefore I can write t1 by t2 is equal how much? 450 divided by 450 <unk>3 450 gets cancel your answer is this one okay your correct answer is this one understood so I applied lamb's theorem now let us apply lame them here also okay so what will you do the simple thing what you have to do is you can just draw one dotted line like this understood or not you can draw a dotted line okay or else you can do one more thing if this is 60 what is the remaining not 30 this is 180 no this This is going to be 120.

[00:11:58]Okay, that is 120. If I draw one horizontal dotted line, if this is 60, is it not 60? And this is not is it not 90? Okay, so this is 90. 90 + 60 is how much? 150 + 150 is >> means this is going to be 60. Okay, direct you know. Is it okay? Okay. So here downward what is the force acting?

[00:12:22]/ which is equal to t1 / sin the angle opposite sin 60 which is equal to ts2 / the angle how much again sin 150 okay now can't you find out t1 and t2 from that so what will you do now anything you can find out you just compare these two first okay after compare these two first these two next and you get the answer so let us write now first t1 by sin 60 <unk>3x2 2 which is equal to sin 150 sin 180 - 1 1x2 sin 60 is 1x roo<unk>3 sin 20 is sin 150 is 1x2 so here 2 gets cancel so t_1 is equal to 20 <unk>3 so t2 by sin 30 sin 150 yes you can do it good idea now it's very easy now t1 is going to be sin 60 <unk>3 by 2 20<unk>3 both are equal okay next you can just see A what is this? A is nothing but 3 I + 4 J. Okay. So what is A? A is equal to 3 I + 4 J. What is B is nothing but 7 I + 24 J here. Okay. Then what else? Find the vector having the same magnitude of B.

[00:13:37]First B magnitude square roo<unk> of 7 square + 24 square which is nothing but 25 units. So I got B value B the magnitude. to find the vector having the same magnitude of B and par to A.

[00:13:56]>> So you have to find out the unit vector of that. If you find out the unit vector of A and multiply with this, you'll get the answer. So unit vector of A means what? A by magnitude of A gives you the unit vector. So APA is nothing but A is how much? 3 I + 4 J. It's going to be five. Magnitude square root of 3 square + 4 square is 5. Is it? Okay. So you got this one. Now what will you do? If you multiply that 25 with this then you'll get a Okay. So what is the resultant? So resultant vector. Okay. So what we get?

[00:14:29]25 into what is this? 3 I + 4 J / 5. So five times. So 5 3 are 15 plus is your answer. Isn't it? Means what? 3 15 4 5 are 20. Your answer is correct.

[00:14:46]That means it is parallel. Okay. Next.

[00:14:50]Find the resultant of a vector shown in the figure. Okay. Complex solve. Okay.

[00:14:55]So what they will do you know I vector J vector that is 5 cos 37 I plus 5 sin 37 J. Okay. Plus again 3 I + 4 J. But I have got a simple method. We not do all this nuisance and nonsense. The simple thing is you have got a common sense here. Now this is five. Okay. What is this? This is 37. And what is this? This is 53. Okay. To exactly 37 opposite three. Smallest angle, smallest value three. And this is 53° because this is 90 means the sum of these two should be 90. So this is three means what is the remaining one? This is four. Okay, that means this is three. So the component E direction three. Then the component direction four. So already E direction three only four is there. Isn't it? So totally this is going to be seven. This three this four is it okay and this is in this direction four is there and three is there this is going to be seven so your resultant is how much your resultant is 7 <unk>2 7 <unk>2 and what is the direction of the resultant 45° this is the result I know so the resultant he's asking okay the resultant of the two vectors is going to be how much 7 <unk>2 and the direction of the resultant is theta which is going to be 45° okay because if opposite side see in any vector If opposite side and adjacent sides are going to be same that means angle is going to be 45° tan 45 7 by 7 is 1. So this is your answer now. So according to your wish you told graph graphical question you asked.

[00:16:29]So how do you solve this graphical question just here. So first of all I these forces are on a body figure they have resultant force only along the y direction. The magnitude of the minimum additional force. Okay it is having resultant force along the y direction.

[00:16:44]That means along the x direction no resultant force zero. It is having resultant force along the y direction means along the x-axis resultant force is not there. Zero balancing condition is there. So what will you write now?

[00:16:56]See in this direction you write 1 cos 60 plus this is how much we'll take now this is 30. This is 60. No because it's a straight length vertically opposite angles are same. So plus 2 cos 60 direction component xaxis 1 60 component xaxis the correct 60 2 sin 30 right side component e direction 4 sin 30 so what are the left side forces 4 into sin 30 1x2 is 2 right side forces 1 into cos 1x2 is right side plus 2 into cos 60 2 into 1x2. Okay. So what is there in right side? Half + 1. So left side of say 2 and right side is how much? 1.5.

[00:17:46]Understood or not? What is the resultant now? 0.5. Difference is going to be the resultant. So how much extra additional force the magnitude of minimum additional force needed is how much?

[00:17:55]0.5. That's it. Is it clear? Understood?

[00:18:03]1x2 1 60 1x 2 2 60 2 into 1x2 and half + 1/ half is 0.5 1 + 0.5 is 1.5 >> next you just see here the sum of the two unit vectors is a unit vector the sum of the two unit vectors is a unit vector then the magnitude of the difference is how much so how will you do this one so you know what is the resultant r² is equal So a² + b² + 2 a cos theta. We don't know what is theta but you know this is 1 and this is 1 square is 1 only know + 2 into 1 because a is 1 here a = b = r= 1 according to the problem that is sum. Okay. So what are we going to get now? 2 cos theta here. So this is 1 = 2 + 2 cos theta. So 2 cos theta is equal to -1. So cos theta is equal to minus 1x2 which is nothing but cos 120°. That means if you take one vector and the other vector this is going to be how much? 120°. Understood?

[00:19:11]If that is 120° then if you make a difference it's going to be remaining angle how much? 60° difference addition angle how much? If the these are the two vectors a bar and b bar and this is minus a in between these two angles angle is 60 120 but difference means you have to make difference of this a you'll get left side so what is the resultant now how will you find out the resultant here now just see the resultant is going to be r equ= square root of 1 square + 2 into cos 60 is what 1x2 gets cancel so r is equal to 1 + 1 which is nothing but roo<unk>3 is your answer just angle will change addition is 120 subtraction means 60 I told you that subtraction is nothing but addition of one positive vector with a negative vector mean this is going to be B plus of minus A and this is going to be V plus A is it clear what I'm saying any doubt is there that's it >> the sum of two unit vectors is a unit vector in terms of vectors not in terms of scalas unit vector is a scale vector vectors It's not a scalar addition, not an addition. It's a vector addition. Okay.

[00:20:24]So here you just see a truck is traveling towards north like this with a speed of 20 m/s and turns due west and travels with the same speed like this 20 m/s. You all know that the resultant velocity is going to be 20 <unk>2. But direction you'll get confusion regarding direction. How do you find out the correct direction? See truck is going in this direction with this speed and again changing like this. So you just see so in this direction and in this direction means resultant will be how much pythagoras theorem. So triangle you say this is going to be the resultant and what is the pythag we going to apply 20 <unk>2 what is the correct direction southwest how did you tell then correct solution see by using parla if you see the parla parla means what two vectors should be away from one point this is okay from one point but that is towards one point that should be changed Here this is okay. This is the correct position. But this is moving towards the tail. But according to Carlo, two vectors should be away from one point. Then only resultant will be there. So this is in the wrong direction. Change the direction. So if you change the direction, this is going to be the real resultant which is this is west and this is south. So southwest and 20 <unk>2.

[00:21:46]Okay. So D option is the correct option.

[00:21:48]In case of whatever you draw, I'll show the resultant direction.

[00:22:07]thinking and okay answer understood. Yes, it's okay.

[00:22:37]Next boat concept man and ideics >> graphs. So graphs you want in chyntics.

[00:22:48]Okay. So graphs I think you learned how to derive equations of motion in chynatics. But you should know which graph represents what isn't it? Okay. So first of all we shall discuss about what is the meaning of position time graph.

[00:23:02]Okay. or this is also called as displacement time graph. So what are these g graphs representing? Let us see here. Okay, I'll give an example. So this is position means xt graph and position time graph isn't it? This is called as position time graph. You only tell me that what is this graph representing now?

[00:23:22]>> Yes.

[00:23:22]>> Yes. So many people will get confusion here. when it is rest when it is constant velocity when it is having increase in velocity which is having acceleration which is having deceleration for that I have got a simple technique so in a position time graph if you want to find out velocity you have to find out the slope isn't it tan theta so now the graph is like this this is position time graph or this is also called as displacement time graph so this is going to be the graph where this is angle theta okay so opposite side is displacement and adjacent side is time. So whenever you want to find out the velocity order at tan theta because opposite side is displacement adjacent side is time according to the general formula also. So you you have to check what is the correct option here by seeing angle theta. So by making by keeping this in your mind here you can decide many situations you can just see I'll tell you many examples like this.

[00:24:17]So how do you decide that this is going to be rest because theta is going to be zero here isn't it? So v is equal to tan 0 is nothing but zero. That means velocity is zero. See time is passing but there is no change in the position like dead body. Understood? So what is the next graph? What is this graph? This is velocity position time graph. Okay.

[00:24:37]Tell me what is this graph representing?

[00:24:40]>> See time is going to be constant and displacement is changing. It is called as impossible graph. See that means what is this mean here? This is 90.

[00:24:49]So velocity is equal to tan which is going to be infinity which is not possible within time you reached from change position it's not possible actually impossible graph okay so next thing is if you for example you draw like this okay and this is position and this is time. Sorry this is time graph. If the graph is like this what is it represented?

[00:25:25]>> See velocity I told you there is only one clue to understand this one. See if you observe theta is going on increasing am or not. Theta is increasing or not. See tan actually but it is greater than 180. So this is negative slope. So negative slope means what does it represent? Negative velocity means body is coming towards you. Train is approaching the station where you are standing negative velocity. Understood? And the two if it is and the two this is uniform velocity.

[00:25:57]It is coming towards you with constant velocity. Is it clear what I'm saying?

[00:26:01]Now so next we shall discuss about what is the velocity time graph. Already have explained this one. Did I explain this one or yesterday? I have proved by using graphs.

[00:26:11]>> So what is this actually velocity time graph? Okay. So if I draw a line here, this is going to be opposite side. Okay.

[00:26:19]This is opposite. This is adjacent. This is time and this is velocity. So acceleration is nothing but change in velocity by time. So that means if you want to find out an acceleration, you have to find out the slope which is opposite side by adjent side. You know acceleration is nothing but change in velocity by time. Let you write like this also which is delta v by delta acceleration. Okay. So if you want to find out the displacement in the case of this graph in this graph if you want to find out displacement what will you do?

[00:26:47]You have to find out area uh area of the graph gives you displacement and the slope of the graph gives you acceleration. Understood? Okay. For example, now see if this is the case now this is a BT graph. So this is a graph which is where velocity is going to be constant. Why velocity is going to be constant? Acceleration is equal to tan 0 which is equal to z. Acceleration is zero means velocity is uniform. Velocity is constant. If I ask to convert this graph vt graph into position time graph.

[00:27:18]So this is displacement time graph or position time graph. How will you convert this vt graph into position time graph? That's it like this. Understood?

[00:27:27]Clear? Okay. So next thing is if I'm going to consider the graph like this.

[00:27:33]Okay. I'm drawing a line like this here.

[00:27:35]Just now what did I explain? No horizontal line. So what is this? This is angle theta. This is velocity and this is time. So what does this graph represent? So acceleration is equal to tan theta. Here theta is constant. Here also this is the clue you have to remember.

[00:27:56]So theta is constant. Aelation is conant mean uniform acceleration case here.

[00:28:03]Okay. So now for example the graph is like this. So this is velocity time graph. Okay. Acceleration is nothing but tan theta. Here theta if you see at going on increasing. So theta is increasing acceleration is also increasing. It's a case of positive acceleration is increasing. Understand what will you write down for that here theta is increasing mean acceleration is increasing. How will you write that?

[00:28:27]Think and tell me now. So what is this?

[00:28:28]What is this? If you draw a horizontal line theta is going on increasing. So acceleration is tant theta is decreasing. So velocity time graph acceleration is also decreasing means deceleration retardation understood or not. Okay. So these are some of the graphs which you have to remember.

[00:28:44]Tomorrow we'll do some problems also based on graphs. Okay. So these are only basics of graphs. Tomorrow we'll do problems. Understand? So tomorrow you know there is a chance of there is reit exam. You know what that >> paper has been need so tomorrow onwards classes will be there in the morning from 7 to 9 like that for need students in that we will inform you when is physics and chemistry and math okay so tomorrow evening we'll send you the time table No.