Let's start. So continuing from the last class everyone, we will start with some very simple questions to begin with. You guys have to check what happened in the last class.
First question, question 1. Today we will solve DPP on this in the afternoon guys. Question 1. A vector is made. from 0 comma 2 comma 0 to 4 comma 2 comma 8 find the vector also calculate its magnitude and direction every time we ask for direction you have to find out the angle with that it makes for the x-axis y-axis and the z-axis so let's use this and make a vector quickly. How did we make last class vector?
First tell me what we will name the vector according to you. arrow but you tell me what I should take in this case take A ok let's take A vector now what we will write here first comes something i cap something j cap and something k cap so what should I fill here go and check what we have written in vector fast x coordinate so this time how much is x final? 4 and initial?
0 so how much will be this? 4 minus 0 0 so we will write this as 4icap did you understand? next what will happen?
then y coordinate means 2 minus 2 which is 0 0 so this is 0j cap then? then z coordinate means 8 minus 0 is 8. 8k. Okay. So, your vector is ready. Look guys.
Now, if I have made the vector, then remove the magnitude and the direction. Come on, quickly. Did you understand this?
Make this quickly. Nishant. Last class you saw properly. Yes sir.
So you can follow this today. Yes sir. Ok continue. As soon as your vector is ready, you can make the magnitude quickly. Therefore the magnitude over here, if everyone remembers.
4 square is 16, 0 square is 0, 8 square is 64. So, this is the root of 80. You can write this as 4 root 5. 16 5 size 80. Okay. So, as soon as it is ready, what we will do is we will draw an angle. So, first of all, it will be cos theta x.
Cos theta x is the coefficient of i, which is 4. divided by 4 root 5 so this is 1 by root 5 okay unfortunately i don't have any such cos theta whose value is 1 by root 5 so i told you in such case we write theta x cos inverse of 1 by root 5 this is theta x Then we move towards cos theta y. Cos theta y will give you 0 divided by 4 root 5. Coefficient of j divided by the magnitude. So this will give you still for 0. When cos theta 0 happens?
When the angle is 90 degrees. The last one, cos theta z. What will be cos theta z?
If we want to make cos theta z, we will have to find out the coefficient of z, which is 8 divided by 4 root 5. This is 2 by root 5. And therefore, theta z will become cos inverse of 2 by root 5. okay you got all all the four answers okay okay the same question The position of rectangular coordinates of a vector is given as 2 comma 3 comma 4. This is given. Okay. So, now, as they said like this, position of rectangular coordinates of a vector is this. You can directly take this change in x. This is change in x.
You don't need to take two coordinates in this. They have already given change. Okay.
So, now, take out its vector, take out its magnitude, take out its direction. find the vector its magnitude and direction okay so this bar without thinking much you can directly write down the vector b bar plus 3j plus 4k you can write 2i plus 3j plus 4k if vector is ready then we will make magnitude this is going to be 2 square which is 4 plus 3 square which is 9 and 4 square which is 16 so this will be root of 29 29 unfortunately is a prime number you will not be able to further simplify it. So if you take out theta x, it will become cos inverse of 2 by root 29. Similarly, theta y will become cos inverse of 3 by root 29. And the last one is theta z, which is cos inverse of 4 by root 29. these are the values everyone is okay with this given c bar equals 2i plus 3j okay given c bar is 2i plus 3j find a unit vector in direction of c bar also find a vector in direction of c bar having magnitude of five units this is the question you can find only unit vector for me I will solve the rest of the question. Will everyone come to make a unit vector?
Just make a unit vector ready. I'm assuming you some lonely unit vector but not a ready car I just question if they seek up up because that's the unit vector is C vector divided by the magnitude of C vector to see vector get back to I plus 3g look is a divide carrying it I'm not so divide carrying a magnitude of CK Magnitude of C is 2 square root of 4. 3 square root of 9. If you add, then 13 is square root of 13. Is everyone okay with this? Now, I will explain you the diagram.
I am telling you what the question is. Everyone, look carefully. You have been asked the actual question.
You have been given C bar. Tell me a vector which is in the direction of C bar, but its magnitude is 5. So, what did I do? I said, if this is C bar, So I have made its unit vector. Now what is the quality of this vector?
The magnitude of this vector is 1. If I multiply this with 5, then I will get a vector whose magnitude is 5 and it is in this direction. Do you understand what I have done? Look at this carefully.
This is my C bar which you gave. What did I make with it? I made it C kappa.
And if I want to make a new vector which is d bar and the value of d bar is 5 times c kappa. Why? Because 5 times c kappa will be 5 times the magnitude is 1. So this vector is 5 times of the original.
Did you understand what I have done? This is how the questions will come. I have given you an intermediate step but they will not give you that.
What they will give you? I have given you this. So tell me a vector whose magnitude is 5 and it is in the direction of c. So first of all you will make that vector a unit. And as soon as it becomes a unit, multiply it by 5 and the work is done.
Everyone is ok with that? You understood the story? Ok. So what will be the d vector this time? Your d vector will become 5 times c kappa.
Therefore you can directly multiply 5 to this. Unfortunately division of root 13 will give you root 13 itself because that's again a prime number 10 i plus 15 j whole divided by root 13 10 by root 13 i cap plus 15 by root 13 j cap all okay Question 4. a bar equals 30i plus 40j. Find a vector opposite to a bar. Sir Tell me Sir, opposite means in direction Yes, try it Shaurya, it's a good question Hmm Yes Pandey, tell me.
Hello. Hello. done sir okay so we can solve this this way most of you guys have the answer a kappa a bar whole divided by the magnitude of a this is 30i plus 40 j magnitude so divide carrying it the magnitude you have a 50 so this gives you if you've seen this this will give you 0.6 I cap plus 0.4 j cap okay make quick essa vector banana just come magnitude does have or who opposite hey a vector say America be bark essa banana maybe bar is sorry this is 0.8 sorry my bad this is 0.8 so my b vector will be minus 10 this will give you minus 10 into 0.6 i plus 0.8 j so it will be minus 6 i minus 8 j you can write it like this did you understand everyone Everyone's okay with this? Chalo, agla sawal. Next question everyone.
Question 5. Is bar tumko diya hua hai, A bar equals 10i plus 20j b bar equals 20i plus 20j. I have given you these two vectors. The question is, find, probably you can just solve and give it to me. find a bar plus b bar second unit vector in direction of a bar plus b bar okay Okay, I have the answer from Shaurya which is very good. You did it very fast.
Rest of the people, hurry up. This is a straightforward addition of A bar plus B bar. If you add it straight, what will you get here?
You will get this as 30i plus 40j. You need a unit vector which is in its direction. So you already have a vector 30i plus 40j.
But you have to divide it by its magnitude which is 50. So this gives you 3 by 5i plus 4 by 5j. If you write it in decimal then also it is correct. Answer is not incorrect. Everyone is ok with this?
Everyone is ok with this? Everyone is ok with this? Everyone is ok with this? Yes sir.
Next. let's go ahead the next question question six i'm actually given a bar equals 3i plus 4j plus 5k find the angle it makes with z axis find the angle that it makes with z axis second what is the y component of this vector equation Whenever they ask you about its y component, x component, z component, they are nothing but coefficients of i, j. So the second answer is straightforward. j's coefficient is 4 because that is supposed to be y component.
So your y component is 4. You do the first part for me. That is more important. That's correct Driti. Darshan, answer is not coming. You can answer any question.
so uncorrect correct sir type if you find this out you get 9 plus 16. plus 25. so if you solve this you get root of 50 which is 5 root 2. you have asked for the x axis and z axis so you don't have to find all the angles. so you have cos theta z which is the coefficient of z which is 5 divided by 5 root 2. so here 1 by root 2 so here clearly theta z will be there. 45 degrees or y component to many but the y a belly y component k out there coefficient of j z component coefficient of k and likewise coefficient of y is called as the x component everyone's okay with that okay up next question question seven Sir, I have a doubt in the fifth question.
Tell me. Sir, we have learned to add vectors in the same direction. Correct.
So, we have learned to add vector a plus vector b directly. that is very good now listen, I can add vectors but I can't add the magnitude of vectors ok, that's it did you understand? yes sir I am glad that you asked this question many people did not ask this question how did we do this?
how is this even possible? you took one vector and added another vector you didn't even teach this but you did it straight forward but some somebody asked me which is very good that you're checking what we are doing question seven look at this the question that they are asking over here is find cos square theta x plus cos square theta y plus cos square theta z This is a Maths question, but it is a very good problem. It is a question that has been asked many times.
Very good everyone. Many people are getting good answers. Shaurya, answer it.
It's easy. I got it. Okay, good.
It's very long. No, it's not long. It's very simple. If it's long, it's wrong.
First, answer individually. First thing, if everyone has answered it carefully, then what will be cos theta x? It will be A1. upon under root of a1 square plus a2 square plus a3 square similarly if you take cos theta y so this will be a2 whole divided by under root of a1 square plus a2 square plus a3 square similarly if you take cos theta z this will be a3 upon under root of a1 square plus a2 square plus a3 square okay what is cos square theta x plus cos square theta y plus cos square theta z it is equal to e1 square upon e1 square plus e2 square plus e3 square plus e2 square upon e1 square plus e2 square plus e3 square plus e3 square upon e1 square plus e2 square plus e3 square.
Okay. If you add all these, you will get this answer 1 directly. Are we fine with this?
This is a fact that you need to remember. It's a fact. Mark it there.
Fact to remember. That direction cosines of squares, if you add cos2θx, cos2θy, cos2θz, it will always give you 1. Is that okay? Sir, what did you say? Say it again. I said...
There is a fact to be marked. Square of direction cosines. Direction cosines is x square. Direction cosines is y square.
Direction cosines is z square. If you add them, it will always give 1. If this is true, then you have to find this too. Find Let's see how you do this.
This should be done now. It will be fun. Nobody has given the correct answer yet.
No, I am sure it is wrong. At least try a little. No, Yash.
Correct. Very good, Sohan. No, Heet. It's wrong. First of all, do you have an idea of what Cosquare is giving?
No, Aarya. Let's do this. Everyone, look.
I always know this identity sin2θ plus cos2θ is 1 everyone knows this so sin2θ can be written as 1 minus cos2θ so this means that I can write sin2θx as 1 minus cos2θx plus 1 minus cos2θy plus one minus cos theta z so we have made this so write what is 1 plus 1 plus 1 3 is happening minus cos theta x plus cos theta y plus cos theta z and therefore this is 2 What have you made, Driti? What is the answer? The right thing to do. Okay. Everyone's okay with this?
You understand, Pranay? Okay? Yes.
Alright. siley following right so now that we're done with some questions we can go slightly ahead with the next thing that we have to do today so the next thing that we're doing today is called as addition slash subtraction of vectors addition or subtraction of vectors just give me a minute guys okay guys right down below that when two vectors when two vectors are added or subtracted or subtracted it always okay is called as the resultant vector this vector is called as the resultant vector okay okay Let c bar be the resultant vector of a bar and b bar. Let c vector be the resultant vector of a bar and b bar.
Therefore, c bar becomes a bar plus b bar. Is everyone okay with this? So if I call c the resultant of a vector and b vector, then c vector will always be a bar plus b bar.
Okay? If I want a bar minus b bar, then what will I do? I will say let d vector be the resultant of a bar. and minus b bar therefore d vector what will happen?
a bar minus b bar everyone is ok with this? because if you know b bar then you must know minus b bar also because if I have written b bar as 2i plus 10j so minus b bar obviously minus 2i minus 10j everyone is fine till here so this is very straightforward it does not require any new idea or logic okay now with this you can do basic straightforward questions question number one given a bar equals 3i plus 2j plus 3k and b bar equals 2i plus j plus 2k. The question is find e bar plus b bar list the domain to you salt and ergo number one e bar plus b bar but not the dough was case at south magnitude of a bar plus b bar banana fair second a bar minus 2b bar and the fourth is magnitude make it a bar minus 2b bar ok, let's do this Thank you. Hi Yash, tell me. Sir, I asked about both.
Both? I can't hear you Yash. Your voice is cut off. Tell me. We can't hear you clearly.
Can you hear him? No sir. Yes. I cut it off in the middle.
Yes. okay so we have five i plus three j plus five k you can write it like this you can call this as C bar so the value of C bar is this so you can write magnitude of C bar or magnitude of A bar plus B bar The answer will be root of 25 plus 9 plus 25. This is root of 59. Similarly, if you talk about D bar, then we are writing D bar as A bar minus 2B bar. So, you have to multiply it before 2 and then subtract it.
So, 3I minus 2I will give you minus I. 2i minus 2j is 0j or similarly 3k minus 2k is minus 4k is minus k so this is your d vector so therefore if you are taking out the magnitude of d vector or you are taking out the magnitude of a bar minus 2 b bar so this will be under root of one negative one square is one zero is zero and this over here is one so this is root of two Everyone's okay? Getting a hang of this now?
Yes, sir. Okay, let's go. Question 2. Sir.
Yes? Sir, my mobile battery... Just a second, just a second. Okay, so we're here. we have a bar plus b bar a bar and b bar is given to you find the angle made by the resultant vector also find a vector which is a negative vector of the resultant solve it Yes, yes, yes, all the triangles.
Okay, so we have some answers over here. let's go step by step so clearly first of all make a vector here whose name is C bar you can call C bar A bar plus B bar if you add these two you will get 4i plus 0j plus 2k so magnitude of C bar equals 16 plus 0 plus 4 this is root 20 this is 2 root 5. So we have prepared the magnitude. Yes. Sir, 4k and 0j will be correct.
Yes, 0j. I just wrote it wrong. 0j. And it will be 4k.
And this will be 4k. Correct. This is also a mistake. Sir, it will be 4k. 3k plus 4k.
Yes, yes. Sorry. So this is 16. This is 16. therefore the answer will change to root 32 this is root of 32 and this will give you 4 root 2 okay so you have to take theta x because theta x is coefficient of i by coefficient total magnitude it is 1 by root 2 therefore theta x will become 45 degrees cos theta y equals zero Therefore, theta y becomes 90 degrees. And similarly, cos theta z, if you make it, will give you 1 by root 2. Therefore, theta z becomes 45 degrees.
Now, we have all this. After that, what you have to make? You have to make a negative vector of the resultant.
So, say d bar equals minus c bar. So, what was c bar? This was its negative. This will go minus 4i.
minus 4k everyone has this yes sir okay next question question An object moves 50 meters north, 30 meters east, and 10 meters south. Find the resultant distance. placement so you first of all solve this question for me and give me the answer in this case I am going to tell you a very good method from vector solve this and get me the answer I already have an answer from Darshan Yash is wrong, Shaurya is correct Yes, so most of you have the answer correct.
Everyone see how I am drawing it. Please stop writing. So currently I am drawing it and telling you.
Then I will solve and show it to you. I have this going to the north. This went 50 meters.
Then from here it is going to the east. This is going over here to 30 meters. And then it ultimately came down from here.
10 meters like this okay just a second okay now next you have done till here now you know your displacement ok next what you are supposed to do you are supposed to find out your displacement so we are doing it according to the diagram so what that tells you we know that clearly from my diagram this over here is 30 and this is 40 therefore we can say that this is 50 But you will see that the diagrams are going to be very complicated. Let's see how to get the answer for this. Everyone just see this, okay?
First of all, I write it like this. The first displacement that I have, everyone look carefully, it is from here to here. It is going only in the y-axis.
So I write it as 50j cap. Did everyone understand this? The second displacement that I have, That is 30 iCap Mohit tell me displacement 3 Hello Tell me displacement 3 I have told you this and this 10 K 10?
How can it be K? It is in Y axis only and j it has to be 10 j but over here it is minus 10 j cap it is in the opposite direction guys so i'll repeat again this is 50 j 30 i minus 10 j so again displacement s1 vector plus s2 vector plus s3 vector So this will be 50j cap plus 30i cap minus 10j cap. So this will give you how much?
It will give you 30i cap plus 40j cap. Does everyone know how to make its magnitude? 30 square plus 40 square.
This will give you the answer as 50. Did you understand how it is? so this was question three for addition or subtraction of vector question four given an object moves 10 meters north 10 meters west and 10 meters upwards Find resultant displacement. Try to make this.
Please understand, north and upwards are not the same. I had told you in the last class. Now think about it and make it quickly. It should be straight forward. Yes, yes, that's correct.
Driti mistake kar ye tu. Darshan correct, Shaurya correct, Driti mistake, Sohan correct, Yash correct, Pranay correct, very good. Iska diagram baat me banata ho, lekin karne ka tarika sida sida hai dekho. Displacement 1 vector, 10 j cap. Displacement 2 vector.
10 minus 10 i minus 10 i cap because we are going to the west or displacement three vector here if you make it will be 10 k cap because we are going up don't forget so if someone asks for total displacement then it will be s1 vector s2 vector and s3 vector so here s1 vector is 10 j cap minus 10 i cap plus 10k cap therefore total displacement vector magnitude will be under root of 100 plus 100 plus 100 this is 300 that will give you 10 root 3 meters this is your total displacement see how easily it has given but if you go to draw it then you will die while drawing it see what will happen here So I will try to make it in 3D. This is my plane currently like this. So my body moved to the North.
So assume it went from here to here. This is 10 meters North. And then it goes to the West. so this is how it has moved from here 10 meters to west and then it goes 10 meters upwards like this so what you are finding is that this is the original position and this one is the displacement you can't see it but it is 3 dimensional and it is quite troublesome See your particle has walked here, here and gone up. You have to remove this.
If you remove this, your condition will get worse. But see how easily vector gave. Did you understand?
Write one more question. Question 5. Given the particle is traveling with a velocity of 20i plus 30j meters per second. Find the displacement it makes in 10 seconds.
Find the displacement it makes in 10 seconds. This is very easy. if everyone remembers the relation of displacement and velocity then you have to apply the same thing displacement is velocity into time so just multiply it and you will get displacement also find its magnitude okay so you're supposed to do it this way your displacement vector equals velocity vector into time so your displacement vector is supposed to be velocity vector into time your velocity vector is 20 i plus 30 j therefore your displacement vector becomes 200 i plus 300 j meters this is the magnitude this is 4 10 raised to 4 plus 9 10 raised to 4 which is square root of 13 10 power 2 meters this is your total displacement for this case until you got this Yeah, the first one I got, the second one I did wrong. Okay, you didn't square it properly.
Okay, Keith you got this. Keith, you got this right. Manav?
Yes sir. Alright, Saheeli you want to say something? Sir, I didn't understand the second one. Okay, see how I did it.
You did this right? Yes sir. We will square this. What will be its square? I have written 2x10 power 2. Is this correct?
Yes. What will be its square? 4x10 power 4. Yes.
Similarly, how did I write this? 3x10 power 2. It will become 9x10 power 4. Yes sir. I have added both of them.
If you don't like that, then we can do it like this. but we are not so much manner many cases short cut clear okay so please understand my ticket next next question question six find the acceleration vector and its magnitude Find the acceleration vector and its magnitude if initial and final velocity are 2i-4k and 3j plus 8k ms. respectively so acceleration vector is called to be taken out so what is acceleration? change in velocity by time final velocity minus initial velocity divided by time i have not given time i have to give time in t equals 10 seconds okay some of you guys have the answer already so here see how to write a vector final velocity minus initial velocity divided by time final velocity is 3i plus 8j you have to subtract this with 2i minus 4k you have to divide this by time which is 10. okay so 3i minus 2i is only i 8j 8k okay so i have 8k over here and that is minus of minus k which is plus 12k divided by 10. unless i've made some mistake mistake yes definitely e3j3i so i was just making a mistake over here i don't have any i so this can be written as minus 2 i plus 3 j 8 k plus 4 k will give you plus 12 k divided by 10. this over here is your acceleration vector therefore a vector can be written as minus 0.2 i my plus 0.3 j plus 1.2 k yes most of you have the correct answer let's go ahead question seven given magnitude root of 1.57 meter per second square yes answer it ariana yeah tk question plus 0.4 j plus c k find c.
This is the question. That's it. They didn't give anything else.
Anshul, what is a kappa? 1 unit It should be a unit vector, that is correct So if you know its magnitude, then equate it with magnitude and get C Yes sir Swarnadeep, I am not getting any answer, bro, yours Yes sir, everyone is saying it already. Everyone is saying it already?
What are they saying? Look, you are asking to put the answer in the chat box, you have not given any answer. Nishant, are you understanding what we are saying?
Yes sir. Are you sure? Yes, answers are also coming but it takes more time to type.
That is alright but considering that this is not the last class and you have seen the video, that is why I am asking. Yes sir. You are following right?
Yes. Alright. Darshan, all these answers are same.
So I am assuming that is the answer. Because four people are getting the same answer. Four people cannot be wrong.
This is what I am thinking. But we will solve this. So see how to do it.
So I know that magnitude of A is 1. We can write it like this. 0.2 the whole square plus 0.4 the whole square plus C square. Take square root of both sides Square it sorry This will be 0.2 whole square plus 0.4 whole square plus c square So here 1 is 0.04 and 0.16 plus c square So 1 equals 0.2 plus c square As everyone told c square's value is 0.8 You can write c as root of 0.8 Or someone has given me this answer, this is also correct.
No trouble. Is that okay? So, we will just stop here for now.
Today, we are going to do worksheets on what we have done today and what we have done yesterday. Is that okay? So, I will see you at 2.30 then again. Have a good day everyone.
Get ready for the lecture. bye guys take care thank you sir bye sir