PHYSICS GIFT (GOSALITES INTERMEDIATE FOUNDATION TUNING PROGRAMME) SADAL KONAR SIR - DAY - 5

Added: 2026-05-06

[00:00:03]Good afternoon.

[00:00:06]Good afternoon. Good afternoon.

[00:00:08]Good afternoon.

[00:00:10]Very good afternoon. Very good afternoon.

[00:00:13]Yeah.

[00:00:15]Yeah, I'm audible? Audio video is fine for all of you? Please respond.

[00:00:20]>> [snorts] >> Quick response.

[00:00:22]>> [cough] [clears throat] >> Yeah.

[00:00:29]>> [snorts] >> Yes. Yes.

[00:00:33]Good afternoon, sir. Good afternoon.

[00:00:34]Good afternoon. Very good afternoon to all of you. Very good afternoon. Hello.

[00:00:38]Hi.

[00:00:39]Hi. Hi.

[00:00:41]Okay. So, we are approaching towards the end.

[00:00:45]So, probably this is penultimate lecture. We will have one more lecture after this.

[00:00:50]Yeah.

[00:00:52]And then we'll have few free time for all of you, for you and for me also, and then we'll come back to the campus.

[00:00:58]We'll meet in the campus.

[00:01:00]Lunch? Yes. Yes. We had lunch, I hope.

[00:01:06]Careless whisper.

[00:01:08]>> [laughter] >> Okay.

[00:01:12]Yes. Yes.

[00:01:13]It's okay. It's okay.

[00:01:16]It's okay.

[00:01:17]Yeah.

[00:01:21]When does college start, sir? Very soon.

[00:01:23]Very soon. You will be communicated.

[00:01:24]Don't worry.

[00:01:26]Don't worry. Um I'm good.

[00:01:30]Take it.

[00:01:33]Fine. Fine. Fine. Fine. Okay.

[00:01:37]Let's proceed.

[00:01:38]>> [snorts] [clears throat] >> Let's proceed.

[00:01:42]Right.

[00:01:46]Mostly. Mostly. I think same time frame will be same.

[00:01:52]Thank you.

[00:01:55]Yeah.

[00:01:57]>> [cough] >> Fine. So, >> [clears throat] >> Right.

[00:02:09]Let's start.

[00:02:11]>> [snorts] [sighs] >> Okay.

[00:02:19]Okay.

[00:02:20]Okay.

[00:02:22]The holidays for us it is starting on 5th. I guess. Yeah.

[00:02:34]I'm fine, man. I'm fine.

[00:02:37]From which state? West Bengal.

[00:02:41]I'm from West Bengal. Okay.

[00:02:44]What now? I mean My Aadhaar card and voter card is everything is in South India only.

[00:02:52]So, I am half Andhra, half Bengali guy.

[00:02:58]Yeah.

[00:02:59]So, So, yes. Today we will start dot product.

[00:03:13]>> [snorts and clears throat] >> Okay. So, dot product. So, please know what is dot product.

[00:03:21]Huh? Please listen.

[00:03:23]Yeah. Mostly tomorrow. Tomorrow. Yes.

[00:03:25]Yes. Correctly tomorrow. Mostly I mean we'll see.

[00:03:32]Suppose there are two vectors.

[00:03:36]These are the two vectors.

[00:03:40]This is A vector.

[00:03:42]And this one is a B vector.

[00:03:45]Angle between the vector is taken to be theta.

[00:03:48]So, the first concept we'll learn today is how to take angle between the vectors.

[00:03:54]Easy concept. I think most of you know it, but still I'll revise it once again.

[00:03:58]>> [snorts] >> How to take angle between two vectors?

[00:04:01]See.

[00:04:03]Angle between two vectors >> [clears throat] >> angle between two vectors are taken are taken on tail to tail mode. Angle between two vectors are taken on tail to tail mode. What do you mean by this? What do you mean by this?

[00:04:29]Look at these two vectors.

[00:04:31]For these two vectors, this is the head of A vector. This is also the head of B vector. This is tail of B vector and this is the tail of A vector. So, their two tails are meeting.

[00:04:40]So, wherever the two tails of two vectors are meeting, there we take the angle between the two vectors. Right?

[00:04:47]As an example.

[00:04:50]>> [clears throat] >> I'm taking these two cases.

[00:04:55]These two cases. Two cases I'm taking.

[00:04:58]So, it will give you clarity.

[00:05:00]Like these are the two vectors.

[00:05:06]>> [clears throat] >> This angle is given to be let us say 60°. This is A vector and this is B vector.

[00:05:13]Here, this is A vector.

[00:05:16]This is This is B vector.

[00:05:24]This angle is said to be 150°.

[00:05:28]So, in both the cases I'm asking you what is the angle between the two vectors? How to calculate the angle between the two vectors? How will you do?

[00:05:36]Huh? How will you do? As I said that it has to be always tail to tail. You have to take the tails, two tails. So, look at this.

[00:05:45]>> [clears throat] >> Here here, this is the tail of A vector, head of A vector. But, here we have a head of B vector. Here we have tail of B vector.

[00:05:56]Right? So, what we do? What we do in the case? You remember that class? I told you we can parallelly shift a vector.

[00:06:04]Raise your hand if you remember we can parallelly shift a vector. All of you remember?

[00:06:09]We can parallelly shift a vector. You remember that concept?

[00:06:13]We can parallelly shift a vector. Huh?

[00:06:15]All those will be useful. See, whatever learning we have, no nothing goes in vain. Everything has connected.

[00:06:21]>> [snorts] >> So, without changing the direction. Yes.

[00:06:24]So, you remember that? So, we will do we will shift the vector. We will shift the vector. Shift the vector.

[00:06:32]So, we have shifted.

[00:06:33]So, now this is B vector. This is the head of B vector and this is the tail of B vector. So, tail tail meeting. This is the angle between them. And what is the angle between them? That is 120°.

[00:06:44]Concept clear or not? Everyone respond, please.

[00:06:49]Concept clear?

[00:06:51]Huh?

[00:06:52]Yeah. So, whenever you see the cases head and tail are meeting head and tail are meeting, what you have to do you have to parallelly shift the vector. Parallelly vector shift you will get the answer. Similar case here. Look at this particular case. Here also, this is the tail of B vector. This is the head of B vector. Here is the head of A vector, but tail of A vector.

[00:07:13]Head tail are meeting. So, head tail.

[00:07:16]Once we see head tail, we do not take the angle. It has to be tail tail. So, for that what we will do?

[00:07:22]We will shift this vector. We will shift this vector. This vector will shift. So, it will get shifted.

[00:07:29]Now the vector is shifted. So, here is the head of B vector. Here is the tail of B vector. Tail to tail is this angle.

[00:07:35]180 minus 50. So, this angle is equal to 30°.

[00:07:41]Correct?

[00:07:44]Okay.

[00:07:45]>> [snorts] >> Idea clear? Fine. Done. So, what I'm writing over here >> [cough] >> So, as a note I will write over here.

[00:07:54]>> [clears throat] >> As a note if >> [snorts] >> head is meeting the tail then shift a vector parallelly.

[00:08:13]Shift a vector parallelly can also be written as extending parallelly and take the angle and take the angle tail to tail.

[00:08:30]>> [snorts] >> That's the concept. Take it. Huh?

[00:08:34]So, respond everyone.

[00:08:37]It seems people have very high heavy lunch today.

[00:08:42]Sir, will you tell these notes again in offline class? 100% we will tell you.

[00:08:45]Don't worry.

[00:08:46]For sure we will tell everything we will tell. Don't worry.

[00:08:50]Huh?

[00:08:51]So, Concept clear?

[00:08:53]So, the concept clear?

[00:08:56]Take it. Huh? That repeatedly I say to learn concept. Learn concept. Please don't memorize.

[00:09:01]Don't memorize. Everything is practically very easy over here. You have to get the root of the concept.

[00:09:06]Once you get the root of the concept, things are very easy.

[00:09:09]One more. So, one more example. No problem.

[00:09:16]So, one more example.

[00:09:18]Go.

[00:09:20]Go.

[00:09:21]So, let's say we have two vectors now.

[00:09:27]This is A vector.

[00:09:30]And this one is B vector.

[00:09:33]This angle is given to be 30°.

[00:09:39]Huh?

[00:09:40]Very interesting. Right?

[00:09:42]Very interesting.

[00:09:44]So, you just shift the two vectors parallelly. As a cross up to you.

[00:09:49]Just shift it like this.

[00:09:53]Extend it.

[00:09:56]So, this has become a vector.

[00:10:01]This has become B vector. The same angle is alternative angle. We get 30° only.

[00:10:07]Yeah.

[00:10:08]Easy?

[00:10:09]Job done?

[00:10:12]Job done? Clear now everyone? Shall we move on now?

[00:10:17]Because there is so much to teach today.

[00:10:18]We'll learn so many things today.

[00:10:22]Yeah.

[00:10:26]Yeah.

[00:10:27]Okay. Shall we?

[00:10:30]>> [snorts] >> So now I will take my previous diagram.

[00:10:37]So we are taking two vectors.

[00:10:40]This one is A vector, let us say. And this one is B vector. Angle between A vector and B vector is equal to theta.

[00:10:48]So as per the dot product, what we write, we write A vector dot B vector is equal to magnitude of A into magnitude of B into cos theta.

[00:11:03]Magnitude of A into magnitude of B into cos theta. Don't worry. Why the cos theta is coming? Someone can ask me, "Sir, why not sin theta is there?" I will explain that part also. Why cos theta is there? Why sin theta is not there? I will explain that also. So please hold on. So magnitude of A, magnitude of B into cos theta. So magnitude of A means what?

[00:11:23]Scalar A.

[00:11:24]Magnitude of B means what? Scalar B into cos theta. Actually, cos theta gives us a number.

[00:11:30]Cos 0° 1. Cos 90° 0. Cos 30° root 3 by 2. This gives a number. These all give you a number.

[00:11:39]Uh adjacent No, no, it's not adjacent.

[00:11:41]There is a There is a proper reason behind it. I will tell you.

[00:11:44]So look at the quantity here. Look at the quantity here.

[00:11:48]This all term, they are basically scalars.

[00:11:52]Everything on the right hand side is scalar. So that is why that is why it is called scalar product. So that is why dot product dot product is called scalar product.

[00:12:10]Dot product is called scalar product.

[00:12:19]Okay. This is the reason why dot product is called scalar product. Hold on.

[00:12:22]Please learn patiently. Have some Uh sorry. No, no, no. Please have patience. I'll explain.

[00:12:29]See, like he said, "I told I'll explain why cos is coming." Immediately you are asking, "Why?"

[00:12:33]Please listen.

[00:12:34]Ex- Everything will be explained.

[00:12:36]Yeah.

[00:12:39]So now, A dot B A vector dot B vector is equal to magnitude of A, magnitude of B into cos theta. Magnitude of A is a scalar. Magnitude of B is also a scalar.

[00:12:47]Cos theta is a number. So everything is a scalar on the right hand side. So that reason dot product is called scalar product.

[00:12:53]You have the clarity again. There are some fixed mathematical rules that we can apply for dot product. Next we'll learn about those rules. Okay now? Fixed set of rules. See, mathematics is a very disciplined subject. You cannot do anything here. You have to go as per the direction given there. What is that?

[00:13:12]First one.

[00:13:14]A vector dot B vector can be written as B vector dot A vector. I can change the position.

[00:13:24]I can change the position of A vector and B vector. I cannot do it here everything.

[00:13:28]A vector dot B vector is equal to B vector dot A vector. Where I can write only in dot product. We'll see in case of cross product we can't write this.

[00:13:35]This is not allowed in cross product, but allowed in dot product. What is this called?

[00:13:41]This particular property is called commutative property.

[00:13:46]Start noting down.

[00:13:50]All my dear friends, start noting down.

[00:13:52]If I practice with you guys, so this property is called commutative property.

[00:13:58]This is called commutative property.

[00:14:00]Start noting down.

[00:14:03]>> [clears throat] >> Okay.

[00:14:17]Right? Next.

[00:14:19]Next. [snorts] Let us say there are three vectors, A vector, B vector, and C vector. We can write Please pay attention. A vector dot B vector dot C vector is equal to I can write A vector dot B vector dot C vector. Please note down.

[00:14:41]>> [clears throat] >> B A cos theta. We learned that magnetic line of vector quantity Kahan ja rahe ho bhai tum?

[00:14:53]Please learn from here. There is no need to impress me. I am already impressed.

[00:14:57]Those are all very primary things. If you think that you are saying very high-fi things, they are very easy things. Hold on. Learn properly.

[00:15:11]Yeah. So this is called associative property. Yeah, someone said correctly.

[00:15:15]This property is called associative property.

[00:15:25]This is called associative property.

[00:15:30]Okay. Number three.

[00:15:32]>> [clears throat] >> Number three. Let us come to number three. Number three is A vector dot B vector plus C vector.

[00:15:42]A vector dot B vector plus C vector is equal to A vector dot B vector plus A vector dot C vector.

[00:15:55]>> [clears throat] >> This is called distributive property.

[00:15:59]This property is called distributive property.

[00:16:08]>> [clears throat] >> So three things it is following. Yeah, very good.

[00:16:11]Right, Vaishnavi.

[00:16:13]So three properties it is following.

[00:16:15]I'm sorry.

[00:16:18]Three properties it is following. One is One is commutative property.

[00:16:22]Next is associative property. And the third one is called distributive property.

[00:16:28]Correct?

[00:16:29]Fine? Clear everyone?

[00:16:32]Right?

[00:16:33]Okay.

[00:16:34]Yeah.

[00:16:36]Right.

[00:16:38]Very good. Very good. Aage badhte hain.

[00:16:42]Please pay attention.

[00:16:43]We are taking A vector is equal to A X I cap plus A Y J cap plus A Z K cap.

[00:16:56]B vector is equal to B X I cap B X I cap plus B Y J cap plus B Z K cap.

[00:17:09]Everyone, please pay attention here.

[00:17:11]What was the formula remember?

[00:17:13]A vector dot B vector is equal to A B cos theta. This was the formula. Right.

[00:17:20]If you take If you take two unit vectors If I take two similar unit vectors, two similar unit vectors, like I am writing A cap dot A cap. So A cap and A cap are parallel.

[00:17:37]They are parallel. They are in the same direction. What is the angle between them? 0°.

[00:17:42]So can't I write this is equal to modulus of A cap into modulus of A cap into cos of 0°?

[00:17:50]This is 1 1 1. So A cap dot A cap is equal to how much? 1. So dot product between two similar unit vector is always equal to 1.

[00:18:02]So dot product between two same unit vector is equal to 1. Dot product between two same unit vector is equal to 1.

[00:18:21]>> [clears throat] >> If I do I cap dot I cap or I do J cap dot J cap or I do K cap dot K cap, all these are equal to 1.

[00:18:35]All these are equal to 1. I dot I, J dot J, K dot K, everything is equal to 1.

[00:18:40]All these are equal to 1. Correct? Fine?

[00:18:44]Yeah.

[00:18:46]Then sir, sir, you tell me, sir, if this is equal to 1, that's perfectly fine. I don't have any issue with that.

[00:18:53]Sir, what will happen What will happen if I do I cap dot J cap then?

[00:19:02]If I do I cap dot J cap, what will happen? I cap dot J cap, what will happen? Let us take magnitude of I cap magnitude of J cap cos I and J angle is how much? 90° ka tha?

[00:19:18]The angle between I and J I is along X axis. J is along Y axis. X axis, Y axis, angle is equal to 90°. Perfect answer.

[00:19:26]0. Yes. So this is equal to how much? 0.

[00:19:30]So J dot K 0. K dot I 0. So that means we are writing dot product between dot product between two mutually perpendicular vectors two mutually perpendicular vectors is equal to zero.

[00:19:55]Dot product between two mutually perpendicular vectors is equal to zero.

[00:19:59]Simple thing.

[00:20:01]Okay?

[00:20:03]>> [clears throat] >> So far so good.

[00:20:05]Please respond everyone.

[00:20:08]Yeah?

[00:20:09]Uh magnitude of A cap is one for now.

[00:20:13]Magnitude of A cap, unit vector magnitude is one only.

[00:20:17]Last lecture Last to last lecture, magnitude of unit vector is one. Don't forget that. Previous lesson you should not forget. Magnitude of unit vector is one.

[00:20:27]Okay? Right. Now let us come to here. So what we're trying to calculate, we are trying to calculate here A vector dot B vector.

[00:20:37]>> [clears throat] >> Whenever we are trying to get A vector dot B vector, whenever we are trying to get A vector dot B vector, there which combination you have to take?

[00:20:49]You are supposed to always take I with I, J with J, and K with K. Why? Because all other combinations will give you zero result.

[00:20:59]So I will take this combination.

[00:21:09]I'll take this combination.

[00:21:11]This combination.

[00:21:13]And this combination.

[00:21:16]I will use I with I, J with J, and K with K. Why? Because I dot J will be zero, so there is no benefit in multiplying that.

[00:21:26]J with K will be zero. I with K will be zero. J with K will be zero. So there is no benefit in taking that. So now how much is coming?

[00:21:33]Please respond how much it is coming.

[00:21:36]It will come as AX BX plus AY BY plus AZ BZ.

[00:21:49]Right? How much it is coming? AX BX plus AY BY plus AZ BZ.

[00:22:00]Please confirm everyone. Quick response from everyone. All of you.

[00:22:04]Jaldi. Hurry up.

[00:22:07]Quick response.

[00:22:10]Yeah?

[00:22:11]Right. Right. Right. Right. Correct.

[00:22:15]Correct. So now that was equal to what?

[00:22:18]Magnitude of A, magnitude of B into cos theta. How much is magnitude of A?

[00:22:23]Magnitude of A is equal to under root of AX squared plus AY squared plus AZ squared into under root of BX squared plus BY squared plus BZ squared into cos of theta into cos of theta. So how much cos theta is coming?

[00:22:52]Cos theta is equal to AX BX plus AY BY plus AZ BZ divided by under root of AX squared AY squared AZ squared into BX squared plus BY squared plus BZ squared. This huge looking formula, which is not difficult, actually application is very easy, helps us.

[00:23:27]This is used for what?

[00:23:29]This is used to find to find angle between A vector and B vector. This is used to find the angle between A vector and B vector.

[00:23:46]Yes, I will say. I will say. Don't worry.

[00:23:49]Yeah.

[00:23:50]Please say here.

[00:23:52]>> [snorts] >> So I'm doing dot product of A vector dot B vector. While I'm doing the dot product of A vector and B vector, what I'm saying that we have to only concentrate I with I, J with J, and K with K because all other combinations will give you zero result. I with I will give you one. J with J will give you one. K with K will give you one. So when you are multiplying I dot I one, AX BX, AX BX. J with J one, AY BY, AY BY.

[00:24:20]K with K one, AZ BZ.

[00:24:22]Now A dot B is equal to magnitude of A, magnitude of B into cos theta. How much is magnitude of A? Under root of AX squared plus AY squared plus AZ squared.

[00:24:33]How much is magnitude of B? Under root of BX squared plus BY squared plus BZ squared into cos theta. So this is the left hand side, this is the right hand side. From both I can calculate the value of cos theta.

[00:24:46]And this equation is helpful for why?

[00:24:48]Because if two vectors are provided, this equation will help me to calculate the angle between the two vectors.

[00:24:55]Fair enough?

[00:24:57]We will work out two three examples, you will get perfect clarity.

[00:25:02]Kya ye sabka ho gaya hai?

[00:25:04]Please respond.

[00:25:06]Is it done? Shall we move ahead now?

[00:25:09]>> [snorts] >> Chale? Ja sakte hai aage?

[00:25:13]Hai na?

[00:25:14]Good. Good. Good.

[00:25:16]Theek hai. Chaliye.

[00:25:17]Okay. Okay. Chalo. Chalo.

[00:25:19]Seekh jayenge. Don't worry. Everything you will learn.

[00:25:22]Pakka seekh jayenge. Koi issue nahi hai.

[00:25:24]Chaliye dekhiye.

[00:25:26]Dekhiye to A vector is equal to Please note down everyone. I cap plus J cap plus K cap.

[00:25:34]B vector is equal to I cap plus 2 J cap minus K cap. Please note down.

[00:25:44]>> [clears throat] >> What we are asking here, we are asking calculate calculate angle between A vector and B vector. Note down.

[00:26:01]Calculate angle between A vector and B vector.

[00:26:06]Calculate angle between A vector and B vector.

[00:26:12]Okay.

[00:26:14]Dekhiye.

[00:26:18]Chaliye. How to do? How to do? See here, very simple.

[00:26:21]Very simple. Very very simple. So cos theta cos theta is equal to what we have written?

[00:26:28]AX BX plus AY BY plus AZ BZ by magnitude of A vector into magnitude of B vector. That's what we have written.

[00:26:44]This is the formula.

[00:26:45]Okay? We can do directly also, no problem. You see, I with I, so AX BX is coming one. J with J, so this is coming plus two.

[00:26:54]K with K, this is coming as minus one by under root of This means one squared plus one squared plus one squared.

[00:27:06]Magnitude of B vector means one squared plus two squared plus minus one squared.

[00:27:14]So this plus one minus one gets cancelled.

[00:27:18]How much you are getting here? Two divided by under root of three into This is four. Four plus two six.

[00:27:28]So what is the answer coming? Two by under root of 18.

[00:27:33]Two by under root of 18. So that is equal to >> [clears throat] >> If you want to further split this, we can write two into three root two.

[00:27:46]So cos theta is equal to So cos theta is equal to two by three root two. So theta is equal to how do we write cos inverse of two by three root two.

[00:28:03]This is how we calculate the angle between the two vectors.

[00:28:07]Yeah?

[00:28:09]Yes.

[00:28:12]Correct.

[00:28:15]Theek hai?

[00:28:16]Don't worry. Plenty of examples we will take. So many examples we will take and you will be perfect clarity. Koi issue nahi hai.

[00:28:23]Aur ek example. Chalo next example.

[00:28:25]Likho. Note down.

[00:28:28]Likhiye. Sab log likhiye. I cap plus 2 J cap.

[00:28:33]B vector is equal to minus 3 I cap plus 2 J cap. Come on. Do it.

[00:28:42]Do it. Very simple. Please do.

[00:29:02]Please do.

[00:29:05]Yeah?

[00:29:11]Correct.

[00:29:12]Anyone with the answer?

[00:29:14]Anyone with the answer?

[00:29:17]Chaliye.

[00:29:18]A vector dot B vector. How much it is coming?

[00:29:25]Jaldi boliye.

[00:29:26]I with I minus three. J with J plus four. That is equal to one.

[00:29:34]Right.

[00:29:36]So now one is equal to magnitude of A magnitude of B into cos theta. How much is magnitude of A? Under root of one plus 2 squared into under root of minus 3 whole squared plus 2 squared.

[00:29:54]So, this is coming under root of 5 into this is 9 + 4 that is 13.

[00:30:02]into cos theta So, how much is that? Under root of 65 into cos theta. So, cos theta is coming 1 by under root of 65. How many got it correctly?

[00:30:17]Cos theta is equal to 1 by under root of 65. How many of you got it correctly?

[00:30:21]Yes, very good. Very good. Very good.

[00:30:25]Yes. Yes. Yes. Yes. Absolutely. Very good. So, theta will be equal to cos inverse of 1 by under root of 65.

[00:30:36]Very good. So nice of you.

[00:30:40]Right. Right. Right. Right.

[00:30:42]Okay.

[00:30:44]Yeah.

[00:30:46]Yeah.

[00:30:47]Right? One more example.

[00:30:53]Let us try. Let us try. I got it. Okay.

[00:30:55]So, try it. Don't worry. Everyone will get the correct answer. Please, next one. Please do. Let us say C vector is equal to I cap minus J cap D vector is equal to I cap plus J cap Let's see who can come with a perfect answer. Very easy one. Tricky one.

[00:31:22]Easiest one.

[00:31:26]Angle. Angle. My demand is find angle between C vector and D vector.

[00:31:38]>> [snorts] >> Angle between C vector and D vector.

[00:31:42]Please be careful.

[00:31:44]Please be careful.

[00:31:46]90 degree. Yes. Correct. Very good. 90 degree is the answer.

[00:31:52]Correct. Very good. Very good. Right.

[00:31:55]How? Sir, C vector dot D vector is equal to I with I. It is coming one. J with J.

[00:32:06]It is coming minus one. Zero. I already told you if two vector dot product is equal to zero, then C vector is perpendicular to D vector. So, angle between them will be 90 degree.

[00:32:23]We have already explained here.

[00:32:26]It's clearly written here.

[00:32:30]Dot product between two mutually perpendicular vector is equal to zero.

[00:32:36]Right?

[00:32:37]So, how much dot product came here?

[00:32:39]Zero. So, what is the angle between the vector? 90 degree.

[00:32:44]So, quickly raise your hand. How many got dot product is equal to zero?

[00:32:48]How many of you got dot product is equal to zero?

[00:32:53]Yeah. Very good. So nice. See so many members.

[00:32:56]So many members. Correct. That means you got the idea, right?

[00:33:01]But do remember this thing. Dot product zero means angle will be equal to 90 degree. Ha. Bilkul. Bahut question karenge. Don't worry. We will solve plenty of questions.

[00:33:12]Yes.

[00:33:13]Chaliye. Next one. Next one. Liko. A vector.

[00:33:20]A vector.

[00:33:21]Right. Let us work out. So many questions we will do. Don't worry. Liko.

[00:33:24]A vector is equal to 1 by 2 I cap plus 2 J cap B vector is equal to minus 2 I cap plus 1 by 2 J cap Calculate theta between A vector and B vector. Come on. Fast.

[00:33:52]Yeah. Now people got the idea.

[00:33:55]Yes. So proud of you.

[00:33:57]Very good. Very good. Very good. Now you got the idea. That's it. That's perfect.

[00:34:03]Yes. Absolutely.

[00:34:05]Correct.

[00:34:08]So, A vector dot B vector I with I. It will be minus 2 into 1 by 2. J with J. It will be plus 2 into 1 by 2. 2 2 cancel. 2 2 cancel. Minus 1 plus 1 becomes zero. Whenever A dot B zero, theta is equal to how much? 90 degree.

[00:34:35]Correct now?

[00:34:37]Done? Yes. See. Very good. Very good YouTube people. So nice of you. Great.

[00:34:42]Very good. Very good. Very good.

[00:34:47]Finally idea clear?

[00:34:49]Right? Okay. So, here we are seeing what if A dot B is equal to zero, then A is perpendicular to B. So, the angle between the vector is coming to be how much? It is coming to be 90 degree.

[00:35:01]Clear here?

[00:35:02]Okay.

[00:35:05]Okay. Next thing here.

[00:35:07]Now I'm saying now I'm twisting the idea a little bit. Same concept, but from a different angle. What is that given? It is given that C vector is equal to.

[00:35:16]Please note everyone. Small A I cap plus 2 J cap.

[00:35:22]D vector is equal to 2 I cap minus 3 J cap Please note down the two vectors.

[00:35:30]Quickly.

[00:35:32]C vector is equal to A I cap plus 2 J cap and D vector is equal to 2 I cap minus 3 J cap. Correct? Yeah. So, question given if C vector is perpendicular to D vector, then find the value.

[00:35:57]Then find the value of A. Please do it quickly.

[00:36:37]Correct. Everyone answering. Yeah. Very good. Very good. Very good.

[00:36:43]Very good.

[00:36:46]Yes. Yes. Yes. Yes.

[00:36:48]Yes. Yes. Yes. Absolutely. So nice of you. Very good. Now you got it. So, here what is the idea? Since C vector is perpendicular to D vector, so C vector dot D vector must be equal to zero. C vector dot D vector must be equal to zero. Correct? So, here I with I is giving how much? 2 A.

[00:37:12]J with J. Giving how much? Minus six.

[00:37:16]So, 2 A minus 6 is equal to how much?

[00:37:18]Zero. So, that means A will be equal to how much? 6 divided by 2. That is equal to 3. This is the correct answer.

[00:37:27]How many of you got plus three as the answer? Raise your hand quickly.

[00:37:31]I could see plenty of in YouTube also.

[00:37:33]YouTube also.

[00:37:35]So many members got plus three as the answer. Very good. Yes.

[00:37:40]Yes. Yes. Yes. Yes. Very good. Very good.

[00:37:44]Very good. So nice. So nice. Correct?

[00:37:48]Are we clear idea?

[00:37:49]Okay.

[00:37:51]Idea clear now? Koi doubt nehi abhi?

[00:37:54]Fine everyone?

[00:37:56]Correct?

[00:37:57]Okay.

[00:37:58]Okay.

[00:37:59]Okay. Right. Okay. Chaliye. Chaliye.

[00:38:03]Next. Shall we work out one more question? Ek aur question kare?

[00:38:06]Shall we do one more?

[00:38:08]Ek aur question kare?

[00:38:10]One more?

[00:38:11]Karte hai. Chalo karte hai. Fine. Done.

[00:38:14]Liko.

[00:38:16]A vector is equal to I cap [snorts] minus J cap plus K cap B vector is equal to Please write. I cap minus J cap plus K cap and B vector is equal to 2 I cap plus 2 I cap plus Let us say 2 J cap.

[00:38:42]2 I cap plus 2 J cap Theek hai na?

[00:38:45]>> [snorts] >> Calculate the angle between A vector and B vector. Easiest one.

[00:38:57]Easiest one.

[00:38:59]Easiest one. So, I'm writing two questions side by side.

[00:39:02]This one and another one.

[00:39:06]Another one is A vector is equal to I cap minus J cap plus K cap and B vector is equal to 2 I cap plus J cap Calculate theta between A vector and B vector. Chaliye. Come on.

[00:39:26]Both the questions. Very good.

[00:39:30]Yes. You are responding brilliantly. So nice of you. Fantastic people you are.

[00:39:36]>> [snorts] >> Correct.

[00:39:40]Very good.

[00:39:42]Very good. So this first one this is pretty easy. The moment I do a vector dot b vector is equal to I with I becomes two.

[00:39:52]J with J becomes minus two. It becomes zero. So cut them. A is perpendicular to B vector. So theta is equal to 90 degree.

[00:40:03]Okay clarity theta is equal to 90 degree. Fine. Well done. Next one is what? Next one A vector dot B vector is equal to how much it is coming?

[00:40:18]I dot I means two. Here J dot J that is coming minus one that is equal to one.

[00:40:24]Now how to get the angle? One is equal to magnitude of A vector, magnitude of B vector into cos theta.

[00:40:33]How much is magnitude of A vector?

[00:40:35]Directly I'm writing one plus one plus one root three.

[00:40:39]How much is magnitude of B vector? Two square four plus one that means root five. Cos theta how much it is coming?

[00:40:46]Root 15 into cos theta.

[00:40:49]So cos theta coming how much? One by root 15. So theta is equal to cos inverse of one by root 15.

[00:41:01]Please check your answer everyone.

[00:41:05]Please check your answer.

[00:41:07]Please check your answer.

[00:41:09]Right?

[00:41:12]Correct?

[00:41:14]Yeah?

[00:41:16]Okay. Yes.

[00:41:18]Yes. Yes. Yes. Very good. Very good.

[00:41:21]Very good.

[00:41:23]Very good. Previous one is 90 degree that is a correct answer you got Shalini. That is a perfect answer you got.

[00:41:29]Right? Yes. Yes. Yes. Yes. Yes. Yes.

[00:41:32]Second one these are not tough. These are actually calculative. You have to just get the basic calculation right.

[00:41:38]Correct? So one last ek aur last question karte then I will go to the next topic. One last question let us do.

[00:41:44]Let us do last question. C vector is equal to please try. I cap plus J cap minus K cap.

[00:41:54]D vector is equal to I cap minus J cap plus K cap.

[00:42:00]Calculate theta between C vector and D vector. This is the easiest one. Chalo do it.

[00:42:08]I want answers from everyone.

[00:42:11]Chalo please do.

[00:42:13]Please do.

[00:42:14]Try everyone.

[00:42:21]Chalo calculate karo.

[00:42:23]90? No I don't think that 90 will come.

[00:42:26]Please pay attention. I don't think 90 will come.

[00:42:30]Yeah. Thoda dhyan se karo.

[00:42:35]Please pay a little bit more attention.

[00:42:39]Yeah. 480 degree? Nahi pata nahi. We'll see.

[00:42:44]We'll see.

[00:42:46]Yeah. Chalo let us solve.

[00:42:50]First C vector dot D vector is equal to how much it is coming? I with I that is one. J with J how much is minus one. K with K that is also minus one. So how much it is coming? Minus one.

[00:43:04]Correct. Next step you go na. Why are you deciding from this step? Next step is what? Magnitude of C vector, magnitude of D vector into cos theta is equal to minus one.

[00:43:18]So how much is magnitude of C vector?

[00:43:20]Root three.

[00:43:22]How much is magnitude of D vector? Root three.

[00:43:25]Cos theta is equal to what? Minus one.

[00:43:28]So cos theta is equal to what? Minus one by three.

[00:43:33]So what is the correct answer? Correct answer is theta is equal to cos inverse of minus one by three. How many of you got this answer correctly?

[00:43:45]Raise your hand please.

[00:43:46]Yes here I can see. Very good. Very good. Here people got the correct answer. Very good.

[00:43:52]But but no no that please please please be careful. This is a wrong perception huh.

[00:43:59]It is not like that.

[00:44:01]It is not like that. We can't write it like this. You got it correctly but the correct method of writing like this cos inverse within bracket minus one by three.

[00:44:10]Otherwise you got it correctly.

[00:44:12]Hena?

[00:44:15]Perfect hai?

[00:44:16]Right? Yes. Yes.

[00:44:19]Yes.

[00:44:20]Right. Right. Right. Right.

[00:44:22]Okay. Okay. Okay.

[00:44:25]Fine? Yes. Perfect. Absolutely perfect.

[00:44:28]Absolutely perfect. One more?

[00:44:31]Okay Mallikarjun Swamy one more. One more chaliye one more.

[00:44:35]Karte hai one more. Likho.

[00:44:38]A vector is equal to kar lete hai. Two I cap plus J cap minus K cap.

[00:44:45]B vector is equal to I cap minus J cap plus K cap.

[00:44:50]Calculate theta between A vector and B vector. Please do.

[00:44:59]Please do. Easiest one.

[00:45:02]Yes.

[00:45:04]Yes. Yes. Yes. Yes.

[00:45:08]Correct. Absolutely correct.

[00:45:12]Right. Right. Right. Right. Right.

[00:45:14]Right. Right. Right. Right. Right. Yes.

[00:45:16]Yes. Yes. That's correct. Perfect now.

[00:45:18]Absolutely perfect.

[00:45:19]Yes.

[00:45:22]Right. Very good. 90 degree 90 degree 90 degree so many good answers are coming.

[00:45:26]Very good. So nice of you. Correct. So A vector dot B vector I with I that is two. J with J minus one. K with K minus one. Two minus two becomes zero. So theta is equal to 90 degree.

[00:45:47]Right. Yes.

[00:45:49]Very good. Very good.

[00:45:51]Very good.

[00:45:53]Very good.

[00:45:54]Correct. Correct. Correct.

[00:45:58]Okay.

[00:45:59]Yes.

[00:46:00]Chale?

[00:46:02]Dekho. Finally let us move on. Enough enough. We have done so many examples.

[00:46:07]Itne sare examples ho gaye. I mean how many more examples you are supposed to do? Bahut sare examples ho gaye yaar.

[00:46:12]This is a easier one. Dot product is easier one.

[00:46:15]Cross product is little bit difficult in terms of direction and all that but dot is very easy. Just practice you need.

[00:46:21]Enough.

[00:46:22]Chalo next dekho.

[00:46:25]So next what we are doing?

[00:46:29]These are two vectors that I'm taking.

[00:46:33]This is A vector and this is B vector.

[00:46:38]Correct?

[00:46:39]A vector and B vector and the angle between them is theta. We are saying the angle between them is theta.

[00:46:47]Right? Okay.

[00:46:49]Ha that's what I'm explaining now Nitra.

[00:46:51]You will understand now. Now you will understand. That's what I'm exactly explaining now.

[00:46:55]So dot product is actually taking projection of one vector along another vector. So what I'm taking dot product it basically means taking projection projection of a vector along along another another vector.

[00:47:27]So dot product ka actual meaning kya hai? Ki it is taking projection of one vector along another vector. It is basically projection of one vector along another vector. A A vector hai.

[00:47:39]How much of this A is lying along B? How much of this A is lying along B?

[00:47:45]So for that what I will do I will take the perpendicular here.

[00:47:52]This is 90 degree.

[00:47:56]So how much of A is lying along B? This much of A is lying along B.

[00:48:03]This much.

[00:48:10]This much of A is lying along B. How much of A is that?

[00:48:14]This is we know that is simply A cos theta.

[00:48:19]Then what is the projection you write?

[00:48:21]You write A cos theta into B so that becomes A B cos theta. I hope it is clear now.

[00:48:30]Now you got the clarity how the cos campus is cos is coming.

[00:48:34]Everyone how the cos is coming? It is clear now?

[00:48:39]It is basically due to the projection of one vector along another vector. It is due to the projection of one vector along another vector. It is basically due to the projection of one vector along another vector. Let me give you an example. Ek aur example lete hai. Dhyan se socho.

[00:48:56]Suppose I'm taking I'm taking A vector as A X I cap plus A Y J cap. Let me draw the diagram also so that you get perfect clarity here.

[00:49:12]This is X axis. This is Y axis.

[00:49:16]And here I'm taking one vector as A vector. Ye hamare vector aa gaye. This is A vector.

[00:49:22]This is angle theta. Dhyan se dekhna.

[00:49:25]All of you please pay attention. This is very very important. Right?

[00:49:29]>> [snorts] >> I want to get projection of projection of A vector along X axis.

[00:49:40]along x-axis.

[00:49:43]What I want to get?

[00:49:45]Projection of A vector along x-axis.

[00:49:48]What do you want to get? Projection of A vector along x-axis. So, what we write?

[00:49:53]We write A vector dot I cap. Why? Because Because along x-axis unit vector is I cap only.

[00:50:03]We write A vector dot I cap.

[00:50:06]A vector dot I cap means A x I cap plus A y J cap dot I cap. How much you are getting there? A x I cap dot I cap plus A y J cap dot I cap. J cap dot I cap becomes zero. How much we get?

[00:50:26]A x I cap dot I cap. I cap dot I cap means one. So, what we are getting?

[00:50:33]A x. What is A x?

[00:50:36]This is A x.

[00:50:39]How much is A x actually?

[00:50:42]A cos theta. We got A cos theta.

[00:50:46]That's how we are getting cos theta.

[00:50:52]There is no book which will get you to this particular level.

[00:51:00]Please respond everyone.

[00:51:03]Please.

[00:51:06]Please >> [snorts] >> quick response.

[00:51:16]Correct?

[00:51:17]>> [snorts] >> Yeah?

[00:51:18]Fine.

[00:51:21]Correct? This is we have done. What do we do?

[00:51:24]You do Yes.

[00:51:26]You take projection of projection of A vector along y-axis.

[00:51:39]Projection of A vector along y-axis. You will see a very interesting thing.

[00:51:44]So, A vector dot J cap. I'm writing directly. How much you will get? A y only, no?

[00:51:52]How much you will get? A y only?

[00:51:55]Correct? So, A y is equal to what? A y is equal to A sin theta. Hold on, please. Have patience. Full concentration here. Full concentration towards me. A sin theta.

[00:52:06]Sir, you you said cos will come. Sir, you said cos will come.

[00:52:11]You said cos will come. And you have to sign up yourself. Wait a minute.

[00:52:15]Here theta is the angle with the x-axis, but if I take an angle beta with the y-axis, it's cos answer madam. Yeah, I got A cos beta only.

[00:52:28]So, yeah.

[00:52:30]Correct.

[00:52:32]I got clarity.

[00:52:34]Fine.

[00:52:36]>> [snorts] >> Can I move ahead?

[00:52:39]I'm saying if you take the angle theta with the x-axis, then along y component is coming A sin theta. But if you take an angle with the y-axis which is to be beta, then the answer will be A cos beta only.

[00:52:52]And according to trigonometry, I can keep on going. I can keep on discussing like this. You see, theta plus beta is equal to 90°. Beta is equal to 90° minus theta. So, sin theta is equal to sin of 90° minus beta is equal to cos beta.

[00:53:14]Trigonometry class.

[00:53:16]Basic mathematics.

[00:53:18]If you know mathematics, you can do wonders.

[00:53:23]Anything you can do.

[00:53:28]Right?

[00:53:31]With one last question, I will end the class today.

[00:53:38]What is the question?

[00:53:40]A vector is equal to Please note down.

[00:53:43]A vector is equal to 2 I cap plus J cap.

[00:53:48]B vector is equal to I cap plus J cap. Question is Calculate projection of A vector along B vector.

[00:54:04]This is an exam related question. Very important question. Keeps on coming in NEET and JEE main like exam. Very important question. A vector is equal to 2 I cap plus J cap. B vector is equal to I cap plus J cap. We have to find out projection of A vector along B vector.

[00:54:20]Projection of A vector along B vector.

[00:54:22]All my dear friends, please write the formula. What is the formula? Look at projection of projection of A vector along B vector is equal to A vector dot B cap.

[00:54:43]A vector dot B cap. A vector dot B cap.

[00:54:48]A vector dot B cap. So, that is what? A vector dot B vector by magnitude of B vector.

[00:54:57]Please write down the formula.

[00:55:02]Please write down the formula. Very easy question. We will solve in less than 1 minute only.

[00:55:09]So, now let us write directly. First, find B cap.

[00:55:13]How much is B cap? B cap is I cap plus J cap by magnitude of I cap plus J cap. How much will that come? I cap plus J cap by root two.

[00:55:27]B cap is I cap plus J cap by root two.

[00:55:31]So, here now put it. A vector is equal to 2 I cap plus J cap dot I cap plus J cap by root two. I with I one, so that is two. J with J, that is one divided by root two. So, your answer is three by root two.

[00:55:52]>> [snorts] >> Don't worry. We have done just one question today on this concept.

[00:55:57]Tomorrow's class, I will again start off with one more example from here and then we'll go on.

[00:56:04]>> [snorts] >> Yeah?

[00:56:09]Fine.

[00:56:12]Okay. Please revise all I'm saying you whenever you going you are going to come to the campus either after 7 days or 10 days. Preparation revise all these things. You are getting the lecture note. Please practice at home. Do remember one thing. Be it any class. Be it any lecture. One hour of lecture contemplated with two hours of practice, then the learning will be completed.

[00:56:34]You are watching the lecture on the virtual domain on the internet domain.

[00:56:38]Correct? But learning will not be virtual. If question is given you are taking the screenshot of the solution from the mobile and pasting on the NEET answer copy. You don't know something. The answer has to come from here.

[00:56:51]So, process is very strenuous. You have to practice. Once you practice, you will get through it.

[00:56:56]Take care.

[00:56:57]Right. Yes.

[00:56:59]QR code is here. And specifically today's class practice much because previous classes they are all basic foundation classes. Today's class whatever we have done, these are exam related things. So, practice much.

[00:57:12]Okay. Huge level practice on a day.

[00:57:17]This is your QR code.

[00:57:22]Opening the again I am saying, please you know, I am just a teacher. I just concentrated on my teaching.

[00:57:29]So, between you and me, you, me, and my subject. These three things are important. Regarding all these things, these will be taken care by the management. We have very efficient, very cordial management. The chairmans and himself, the directors, answer and all, they are very careful. They will communicate to you. You will get proper information. Don't worry about that.

[00:57:48]Everything will be communicated. I know you are agitated and anxious enough to come to the campus at the earliest. I don't feel I'm getting there.

[00:57:56]Okay. Hello. Bye everyone. See you tomorrow in the next class. Please practice. Bye.

[01:00:22]>> Mhm.