 So our agenda today would be three-dimensional geometry, 3D geometry, what we call it. Three-dimensional geometry is actually quite a big topic. I mean, if you see from J point of view or from competitive exam point of view, this chapter is quite a lengthy chapter. Okay. Let me give you an overview of the chapter. First of all, what all are there in three-dimensional geometry from your J point of view. So from J point of view, you have the first part is your introduction to 3D geometry, where we are going to talk about our right-handed coordinate system. What's the right-handed coordinate system? Right-handed coordinate system. Okay. We are going to talk about the concept of distance formula. Okay. We are going to talk about section formula and things associated with section formula like coordinates of centroid, etc. These three parts are basically called introduction part. This is called the introduction to 3D geometry. Introduction to 3D geometry. This is what we are going to cover in today's session. Okay. So in grade 11th, you are only supposed to know this much. Right? Yes. Only this much. What is the right-handed coordinate system? What's the distance formula between two points? What is the section formula? Of course, we will be doing a lot of problems in and around this concept. Okay. So the main aspect of 3D geometry comes when you go to class 12th. So in class 12th, you will have the following concepts. One is the concept of direction ratios. Direction ratios and cosines. Okay. And before that, we have to do all the concept of vectors. In class 11th, you only have introduction to 3D geometry, but in class 12th, you have vectors and 3D geometry. So 3D geometry part is studied along with your understanding of vectors. Okay. So vector will be preceding this chapter. Next will be the concept of 3D lines. Okay. And there are a lot of concept in the three lines. I may not be able to list down all of them. For example, you'll be talking about vector equation of 3D line, Cartesian equation of 3D line. We'll be talking about angle between two lines. We'll be talking about distance of a point from a line. We'll be talking about distance between two parallel lines. We'll be talking about distance, shorter distance between two lines, et cetera, et cetera. So there are a lot of concepts, which I will not be stating down here. Okay. But 3D lines itself takes around, you can say one class for sure. Easily one to two classes it'll take. So six, seven hours of subject matter is there just in the line spot. Next is your concept of planes. Okay. So we'll be talking about planes. That's another three dimensional object or figure you can say. So again, with respect to planes, a lot of things will be there, vector equation, Cartesian equation. There are different types of equations, normal form, intercept form. You'll be also talking about interaction of line with the planes. There will be concept of family of planes, asymmetrical form of the equation of lines, et cetera, et cetera. So a lot of concepts are there. Planes itself will take two more classes. So as you can see in class 12th, you have at least four classes required to do 3D geometry, at least, okay. It can exceed that as well. And lastly, we'll also make a special mention of a 3D figure called spheres. Okay. But spheres is just for testing your basic concepts. Normally in JEE, JEE main and advanced questions based on spheres are not asked, but yes, I have seen some questions asked in CET on spheres. So these four topics we will be covering in detail in class 12th. So this will be our subject matter of class 12th, not in class 11th. So after today's session, please don't expect that you'll be able to solve the JEE problems or the past year problems of 3D geometry. In fact, 80% of the questions, 90% of the questions, you will not be able to solve despite attending today's session, because today's session is only on introduction part. Is it fine? Okay. So in your school, have they started with 3D geometry? Has it been done in school? Yes, no, maybe. Okay. Conics anyways, we are also going to start. Okay. So Conics will take around four to five classes. Okay. Okay. So tomorrow's session will be on Conics. Don't worry a formal reminder will be sent to you by the before the end of today's session. Okay. So let's get started with our concept of introduction to 3D geometry, introduction to three-dimensional geometry. So before we talk about anything, it's very important to understand. In three-dimensional geometry, we normally deal with three axes. Okay. Just like in two-dimensional geometry, we had two axes. Right. So if you recall in 2D geometry, what do we have? We have an X axis and the Y axis like this. Correct. And these two are basically two axes, which are perpendicular to each other. Okay. And this plane is called the R-square plane. R-square plane. R-square. R stands for real. Real square means both the coordinates used while, you know, presenting the position of a point are real numbers. That is why they are called R-square. This is called an R-square plane. Okay. Now, in your 2D geometry, it actually doesn't matter which you call as the X axis, which you call as the Y axis, they're just two perpendicular lines. You can call any of them as X axis, any of them as the Y axis. Okay. But normally what do we do? We keep the X axis as the one which basically spans from east to west. And Y axis, we call it as the one which spans from north to south. Okay. But normally there is no such restriction like that. Right. So, the way we represent the coordinate axes in 2D plane is by representing position of any point with respect to these two perpendicular lines. Correct. So, this was in the case of 2D geometry. You already had a bit of exposure in straight lines chapter. Whereas in 3D geometry, everybody please pay attention. In 3D geometry, you need three axes. Okay. You would require three axes. Let me draw them for you. So, let's say these are your three axes. Now, you can all see that I have drawn three axes. Let me name it. I'll be only writing the positive part of it. So, X, Y and Z. So, these are the three axes which are mutually perpendicular to each other. So, this is 90 degree. Okay. This is 90 degree. And this is 90 degree. Okay. So, these are the three axes which are mutually perpendicular to each other. But, any three mutually perpendicular lines, when you are naming it as X, Y, Z, there is a rule that we follow, which is called the right-hand thumb rule. Right? We cannot call any three perpendicular lines to be X, Y, Z in any random fashion. There is a rule. There is a way. There is a, you know, a fixed mechanism by which we name them X, Y, Z. Okay. And that rule is the right-hand thumb rule, which I'm going to explain to you right now. So, we follow a right-handed coordinate system. Why? Because our coordinate system is in such a way that if you extend the fingers of your right hand along the positive X axis. Okay. All the four fingers of your right hand. If you extend it along the positive X axis and curl them naturally towards Y. Curl them naturally towards Y. So, naturally towards Y means this curling. Okay. So, this is a natural curl. Not a unnatural curl. Many of you may be able to curl your finger backwards also. As I have seen some people who can do that, but that's not a natural way of curling the finger. Natural way of curling the finger is like this. Okay. This is called a natural curl. So, naturally when you're curling from X to Y, this thumb actually should give you the direction of the positive Z axis. Are you getting my point? So, this is what we call as a right-handed coordinate system. So, by default in our 3D geometry, we'll be following a right-handed coordinate system unless until stated otherwise. Unless until stated otherwise. So, many students ask me, can it happen that the question center will give that? I know we are following a left-handed coordinate system and then he can give a problem. Yes, why not? In your vectors and all, you have got questions in the past which are based on left-handed coordinate system. Now, what is left-handed coordinate system? Again, a left-handed coordinate system, if you stretch your fingers naturally towards X axis, okay, I hope you can see the camera and curl it towards the Y axis, then your thumb direction, that will be your positive Z axis, right? So, in your left-handed coordinate system, if this four fingers are towards positive X axis and when you're curling it towards Y axis, that means some, I mean, let's say a line coming out from my palm, towards the positive Y axis, then my thumb is towards the positive Z axis, that is left-handed coordinate system, right? So, if a question says, we are following a left-handed coordinate system, then please remember, your I cross J would become, write it down on the chat, minus K exactly is okay. But in the right-handed coordinate system, I cross J is your K cap, like you already have done it in vectors with the SM. Are you getting this point? So, can they ask questions on left-handed coordinate system? Yes, very much. So, be prepared with it. Okay. Now, another way to understand the same thing, another way to understand a right-handed coordinate system is, let's say you are standing in such a way, let's say this is your right hand, okay, and let's say this is your left hand. So, they are making 90 degrees with each other. Okay, right hand, this is your left hand. Then your head will be along the positive Z axis. Are you getting my point? So, let me show myself on the camera. Let's say this is my right hand. This is my right hand, everybody, okay? So, I'm stretching it like this. This is my left hand, I'm stretching it like this. Okay, I hope you can see me. Okay. So, if this is your X axis, this is your Y axis, then your head is the Z axis. Okay, is it fine? Any questions? Any questions, any concerns about a right-handed coordinate system? Clear? So, in right-handed coordinate system, your I cap cross J cap is K cap. Your J cross K is your I. So, basically, we follow a cyclic nature like this. Okay. I cross J is K. J cross K is I. K cross I is J. Okay. And in left-handed coordinate system, your I cross J will become negative K. J cross K will become negative I. I mean, I cap J cap and K cap is the normal, you know, that unit vectors as you already know it. And K cap cross I cap will become negative J cap. So, please remember this while solving questions. Okay. Now, I'll give you a simple situation. Tell me, is it showing a right-handed coordinate system or a left-handed coordinate system? So, identify the coordinate system followed here. Okay. So, basically, it is something like this. This is, let's say, X. Okay. This is, let's say, why? This is, let's say, Z. Okay. Is this a right-handed coordinate system or is it a left-handed coordinate system? Please mention on the chat box. Okay, Vishal. Okay, Arya. Satya, very good. This is not a rocket science. You just have to, you know, follow the mechanism which I discussed with you. Stretch the fingers of your hand along the positive X axis. Curl your fingers naturally towards Y axis. Then see which is the direction of your thumb. If the direction of the thumb matches with your Z axis given to you, then it is a right-handed coordinate system. If it is opposite to the direction, it is left-handed coordinate system. So, what are you getting? Okay. Arya has changed her answer. What about Venkat? Along the X axis, I've stretched all my four fingers. Okay. Now I'm curling it towards Y. Okay. So, my thumb is going towards left. Right? The thumb is going towards left. How are you getting a point? So, basically, when I, when I stretch my right hand, my right hand will be like this. Okay. And now I curl the finger naturally towards Y. So, this is my Z coming out to B. But this is opposite to it. Right. So, we are following. This is an example of a left-handed coordinate system. Okay. Let's say left-handed coordinate system. Okay. So, are we following this type of a system? Unless you're interested, no. We are not following this type of system. So, if you're making XYZ axes, don't by mistake also name them like this because this will become a left-handed coordinate system. Okay. Okay. So, what about this one? I'll give you one more example. Yes. Tell me, is this right-handed coordinate system or is this left-handed coordinate system? Right-handed or left-handed? Very good. Very good, Harshita. Satyam. Right. So, again, if you stretch your fingers naturally towards, so this is how your fingers will be stretched C. Okay. So, now you curl your finger naturally. So, it will be like the C. This is how I'm showing it. From X axis, you're curling it towards Y. Okay. Thumb goes down and does my Z axis direction match with that? Yes. So, it is an example of a right-handed coordinate system. Is it fine? Is it fine? Okay. Now, having discussed about left-handed and right-handed coordinate system, now let us talk about octants. Let us talk about octants. Now, if you recall, in your 2D geometry, in your 2D geometry, we had the 2D space divided into four quadrants. Okay. Because they were four, they were called quadrants. So, this was the first quadrant, second quadrant, third quadrant, fourth quadrant. Right. So, what happened is that your X axis and the Y axis, the two perpendicular reference lines, or you can say the two perpendicular axes, divided the 2D space into four quadrants. Okay. So, they are called four quadrants. And this is the way we basically have these quadrants. You cannot name them in any random order as you want. For example, you cannot start one from here. One is basically this quadrant. So, by convention, we call this as the first quadrant. You cannot debate. You cannot say why this is first. Why not this one is first? Why not this one is first? So, this is by convention chosen to be the first quadrant, second quadrant, third quadrant, fourth quadrant. Right. But this was the story with respect to your 2D space or 2D geometry. But in 3D geometry, since you have, since you have three axes, X, Y, and Z. So, these right-handed coordinate axes divide the 3D space into eight compartments, which we call as octants. Let me show you a diagram for it. Maybe you'll be able to connect to the diagram faster. Okay. So, let me show you a diagram. Okay. If you see on, on your screen, let me just show you, this is your X axis. Okay. I can see the green line on it. Okay. Maybe I'll use some different color. Maybe let me use a red. Yeah. Do you see this line? This is your X axis. Okay. This is your Y axis. This is your Z axis, forming a right-handed coordinate system. And think that, and think that these are like, you know, planes which are passing through two of the axes, you know, at a time. For example, if you see this purple one is basically passing through, I mean, X axis and the Z axis. That means it is containing the X axis. Okay. So, this plane that you see, let me write it down with the green color. This plane that you see, this plane is your XZ plane. Why does it call XZ plane? It is because it is containing the X axis and the Z axis. Okay. Do you see a orangish? I don't know what color you will call it. Orange only, right? Any special name to this color? I don't know. What do you call this? People who are into arts and all will be able to know it. Orange only, right? Okay. Orange only. Fine. So, this is a plane which is basically covering the Y and the Z axis. So, you will call it as a YZ plane. Correct? Because it contains the Y axis and the Z axis. Am I right? And this plane which is the grayish plane, this is your XY plane. So, these three planes, they divide the entire 3D environment. So, those planes remember they are infinitely extending geometrical figures. Plane doesn't mean this is a small sheet like this. Plane, just like a line extends indefinitely, a plane will also extend indefinitely. Okay. So, these three planes, they divide the 3D space, the space, you know, the 3D, into eight compartments, which we call as octents. Okay. Now, our natural question will arise in your mind, sir, which is the first octant, which is the second, which is the third, which is the fourth, which is the fifth, which is the sixth, which is the seventh, which is the eighth. Don't worry. I'll tell you that. So, this is called the first octant. This compartment is called the first octant. Okay. Now, don't worry about the signs as of now. I'll talk about these signs a little later on. Okay. So, this compartment, I hope you are able to see this box. You see this box, which I'm shading with black. Okay. Have you seen cubicles in office? I'm sure most of you would have visited your parents' office and all. And again, even in NPS, they have the new, the new staff room, right? They have cubicles for the teachers, isn't it? Okay. So, think as if these are cubicles. Okay. So, four on top and four in the bottom. Of course, nobody can sit in the bottom. So, these are like four cubicles. So, this cubicle is your first octant. Clear. Similarly, this one is your second octant. This one is your third octant. And this one is your fourth octant. Okay. So, if you see from the top, one, two, three, four. Okay. Now, the one below the first one is called the fifth octant. This is your fifth octant. The one below the second is your sixth octant. I cannot show it because it is away from my view. Okay. The one below the third one is your seventh octant. And the one below your fourth one is your eighth octant. So, altogether, there are eight octants. So, one, two, three, four. Then five, six, seven, eight. Are you getting my point? So, think as if there are four sitting places on top and four, no, you can imagine it as if in a box, if you have made these partitions, the top four partitions are your first, second, third, fourth, and the bottom four partitions are fifth, six, seven, eight. Are you able to visualize it? Everybody's able to visualize it? Huh? Yeah. Good. So, think as if, like, you know, somebody is keeping his laptop here and he's working. Okay. So, let's say he's sitting on in this cubicle. So, his leg will be in the eighth quadrant and his laptop will be in the fourth quadrant. Okay. So, this is how that person will be sitting and working. No, correct? So, his leg will fall in the eighth quadrant and his laptop will be in the fourth quadrant. Got it? You're able to visualize it now? Great. Okay. Now, because of these eight quadrants or because of these three axes, in order to locate the position of a point in 3D space, we need three coordinates, X, Y and Z. Unlike in 2D space, we only require two coordinates, X coordinate and the Y coordinate. In 3D space, we require three coordinates. So, let us first discuss what is the meaning of the three coordinates and what do they actually represent. So, let me just write it down. These are your eight octants. Eight octants. If you want to draw this in all human droid, but I mean, more important to make a visual picture in your mind. Okay. So, now, you drew it almost. Okay. Very good. All right. So, let's talk about the coordinates. So, when you have the three coordinate axes, okay, let's say I draw a right-handed coordinate axis. Okay. Let me call it as X, Y, Z. See, it is not necessary that you have to always make your coordinate axes like this, right? You can name them X, Y, Z in any random fashion you want, but should follow a right-handed coordinate system. Okay. So, please don't think that, okay, since every time sir is making like this, I should also be following this. You can name any one of them as X, Y and Z, but provided it should be in a right-handed fashion. Okay. Anyways. So, first, let us understand what are the planes I've already told you. This is called the X, Y plane. This plane is called the X, Y plane. Okay. This plane, this plane, which I'm shading here, this plane is called the Y, Z plane I've already told you. And one more plane, which is basically in the front of your screen. Okay. This is called the X, Z plane. So, think as if there are walls like this. One wall is, you know, I mean, just look in your room. Okay. I'm sure everybody is sitting in a room. Right. Take one corner of the room to be like the, the meeting point of the, the X axis, Y axis and Z. Okay. Let's call it as the origin. Okay. Now, at the corner of the room, you put the origin. Okay. Let's say this is my X axis. This is my Y axis and Z axis goes up. Okay. Just look at the corner of your room. I imagine that there is an origin kept at the corner. One edge is the X axis, Y axis, and of course making a right-handed coordinate system. Z is your up. Okay. Now the wall where your, the wall which contains your, let's say Z axis and your Y axis, that is called the YZ plane. Correct. The wall that contains the X axis and the Z axis, that will call be, that will be called your XZ plane. And the floor of the room will be your XY plane. That's what I have shown here. Correct. Now, if you want to represent the position of a point, let me again draw it. We'll, we'll keep, you know, needing these 3D system every time. Yeah. Yeah. So let's say I draw a point here. Let's say P point whose coordinates are A, B, C. Then what does this A represent? A basically represents the directed distance from the YZ plane, right down. Directed. Directed means it could be positive or negative, depending whether you are measuring it along the positive X axis direction or negative X axis direction. So from the YZ plane, if you are going in the positive X direction, A will be positive. If you're going in the negative X direction, A will be negative, right? So A is the directed distance from YZ plane. Got it? Okay. B is what? B is the directed distance, directed distance from XZ plane. Okay. And C is the directed distance, directed distance from XY plane. Okay. What does it mean? That means if a point is on the YZ plane, A coordinate will be zero. Okay. So A will be zero if P is on YZ plane. Please remember this. B will be zero if P is on XZ plane. And C will be zero if P is on XY plane. Please remember that. So if a point is on the floor, its Z coordinate will be zero. That means C will be zero. Are you getting my point? If a point is on the YZ plane, that means on this plane, A will be zero. And if a point is on this plane, the plane which is exactly here, its Y coordinate will be zero. Clear? Right? So in order to reach A, B, C, you have to move A units depending upon the sign of A, whether towards right or left, then B units like this and then C units up like this. Then only you will reach this point. So A in the direction of X axis, B in the direction of Y axis and C in the direction of Z axis, then only you will be able to reach this point. Is it clear? Any questions, any concerns? Okay. Now, the sign of A, B, C depending upon the octants are like this. So I'll show you an image. I've, you know, captured an image just to save your time and my time. Okay. I hope you can read this properly. So this is the sign of XYZ in all the eight quadrants. In the first quadrant, you'll see that XYZ are all positive. So any point in the first octant, sorry, if I'm mistaken, I've said a quadrant. Please, you know that because it's a slip of tongue. In the first octant, XYZ sign all will be positive. Please note that. In the second octant, X will be negative, Y and Z will be positive. In the third octant, both X and Y will be negative. Z will be still positive. In the fourth octant, X will be positive, Y will be negative and Z will be still positive. Okay. Now here, how do I remember these signs? See, just remember one thing. Just remember one thing. In the top four octants, Z is always positive. See here, I'm circling it with blue. And your XY sign is same as what you normally have in a 2D quadrant system. In 2D, what happens? First quadrant, X and Y both are positive. So see here, positive, positive. In second quadrant, what happens? X is negative, Y is positive, like this. Third quadrant, what happens? X and Y both are negative. Fourth quadrant, what happens? X is positive, Y is negative. So the way to remember this is in the top four octants, first, second, third, fourth, Z is always positive and X and Y will follow the same way, same sign scheme as your quadrant system following 2D geometry. Clear? Okay. In the last four octants, that is your fifth, sixth, seventh and eighth, you can see your Z is always negative. Negative, negative, negative, negative. Third quadrant, your XY follow the same as what they follow in a quadrant. So in the fifth, both are positive. Sixth, X is negative, Y is positive. Seventh, both are negative. Eight, X is positive, Y is negative. Clear? Easy to remember it. So if I give you a point, will you be able to identify the octant in which it lies? I'm sure you can. This is a super easy chapter, right? Introduction, there is nothing much. Normally you can cover all these things in two hours also. Okay. So normally I keep a class of three hours in case we want some more discussion and all. Okay. Any questions? Any questions, any concerns? Okay. So time for some questions. Let's take some questions. You'll find very super easy questions coming your way. Okay. So I would request you to answer them quickly as we move on. So first question is, for every point on the XY plane, which is true, poll is on. I want everybody to get this right. Yes. Yes. Come on. Please vote. Just nine of you have voted. Okay. Have you got your dates for your final exam? Or is there any semester exam also happening in between? Or directly final term will happen? How is it? UT, UT, no, no, no, don't count UT. I don't count UT at all. UT is like you should be always be prepared. I'm talking about your final exam. The one which you write to go from 11 to 12. Okay. You haven't got your dates, but you're guessing it will be in February. Okay. Thank you. Thank you for letting me know that. Satyam, thanks a lot. One in December, one in January. Okay. I think by December, we should be able to cover almost everything. Thanks to this extra session that you will be having. Okay. Well done. I think let's stop the poll. One of you have responded out of 19. And all of you have got same answer, which is option number C. That is absolutely right. Okay. As I already told you on the XY plane, Z coordinate will be zero. Okay. On the XY plane, Z coordinate will be zero. So the way to remember it, whichever plane you are, right, and whichever alphabet is missing out of XYZ, that coordinate will be zero. For example, if you are on YZ plane, X will be zero. If you are on XZ plane, Y will be zero. Are you getting our point? Next, what is the distance of the point ABC from the X axis? I'm relaunching the poll. What are the distance of, what are the distance of ABC from the X axis? Okay. Satyam, almost one minute, 30 seconds over. See, this is your point. Let's say I put a point here, ABC. Okay. Now in order to reach this point, what do you do? You move A units. Okay. Along X axis. Let's say. Okay. Then B units along Y axis. And then you move C units along Z axis. Right. That's how you end up, you know, so this is how the motion is. This is perpendicular and this is perpendicular. Okay. Now the question is, what is this distance? So the question setter is asking you, what is PM? Will you, we will be able to solve this now. So those have not responded. I can see seven of you have not responded. Now can you respond? A, this is B. This is C. So how much is PM? Everything is there on the diagram. Okay. Five. Four. Three. Two. One. One. Go. Okay. So let me ask somebody. Vibhav, which option you have gone for? Any options? Vibhav K. Current answer. Okay. Now you tell me which option. C. Vibhav C here. From the diagram, it is very obvious. Okay. By the way, most of you have voted for it. That's actually correct. So let me see all of you. Vibhav, please pay attention. In fact, everybody who could not answer this, please pay attention. So from this diagram, it is 90 degree over here. So this P, let me call this point as N, PNM. They are, they form a right angle triangle. So PM square is PN square plus MN square. Okay. So PN is C square. MN is B square. Okay. So PM square is this. So this becomes your distance from the X axis. Okay. So option number B is correct. In fact, please remember that if there is a point ABC, it's distance from the X axis is this. Let me write it like this distance, distance X axis. Okay. What about distance from Y axis? This distance from Y axis will be A square plus C square. This will be your distance from Y axis. What will distance from Z axis? A square plus B square. This will be your distance from Z axis. I think in NPS, your school only NPS, this came as actually a question, objective question in one of the papers. Okay. So please keep this in mind. Any questions, anybody? Any questions, any concerns? Let's take few small, small questions like this only, just to make your basic understanding clear. What is the perpendicular distance of 6, 7, 8 from XY plane? Simple. You should answer this question within five seconds. Five seconds is also too much. By the way, even if they drop the word perpendicular, it is to be read as the same. Okay. Distance and perpendicular distance, they mean the same. Good, good, good. I have got almost 10 people responding. Okay. Okay. Should we stop now in the count of five? Five, four, three, two, one, go. Okay. I'm slightly surprised because even though most of you got it right, nine of you got it right, but six of you made a mistake. Okay. I don't know why people voted for none of these. See, again, this is not a rocket science. Simple imagination. If there's a 0.678. Okay. This is your XY plane. Okay. This plane is your XY plane. Now think as if the floor of your room, that's your XY plane. How far is this point from the floor of your room? Now, in order to reach this point, what did you do? You basically went six units along x-axis. Then seven units along y-axis, and then eight units along z-axis. That's how you reach this point. Yes or no? So six, seven, eight. So how far is this point from your floor of the room? Will it be six or will it be seven or will it be eight? You tell me. Achintya, you tell me. It will be eight. So why did you answer with the option C? So option A only, correct? Six represents the distance. See, whatever coordinate, this represents the distance from y-z plane, not XY plane. This will represent the distance from x-z plane. This will represent the distance from x-y plane. So whichever coordinate you are referring to, just remove that from x-y-z, that plane you are referring to the distance from. For example, this is your x-coordinate. So from x-y-z, remove x. What do you get? Y-z. So you're referring to the distance from y-z plane. That means from this wall. From this wall, you are finding this distance. Is it fine? See, all these are very simple questions, but they are just helping you to visualize the entire situation. Once you started visualizing it, location of a point where is x-y plane, y-z plane, x-z plane, everything will fall in place. That is why this chapter is called introduction. Because in class 12, we will not get a chance to do all these visualizations. There will be directly working on difficult questions. I think I have done the basic ones with respect to, okay, let's take this one. This point lies in which octet. Again, a very simple question. Let me launch the quote. This point lies in which octet. Correct, Satyam. See, again, people are giving wrong answers. Why? 100% should get the right answer. Even one person is making a mistake that hurts me. Okay, good, good, good. Should we wrap this up now? Five, four, three, two, one, go. Yes, right, Arya? The answer is actually option number B. See, as I told you, first look at the z coordinate. Z coordinate is positive. Means you are looking at one, two, three, four. One among one, two, three, four. Now, just look at your x-y. X is negative, y is positive. Means you are in the second option number B. Okay, I'll give you more questions then. Since you have made a mistake here, I'll give you more questions. Okay, tell me three, minus one, minus two, which octet? Minus three, one, minus two, which octet? Three, minus one, two, which octet? Okay, let's do these three. First, second, third. If you want to get number your answer and put the coordinate. Very good, Satyam, Vishal, Neel, Arya, good, good, good. Arnav, very good. Okay, first one, since this is negative, means you are either looking at fifth, sixth, seventh or eighth. Okay, now in your mind, fifth, sixth, seventh, eighth should be like your first, second, third, fourth, in terms of coordinate. Okay, so now x is positive, y is negative. X is positive, y is negative means you have come to five, six, seven, eight. So eighth quadrant, sorry, eighth octet. So this is your eighth octet, octet number eight. Is it fine? Second one, again, since this is negative, again, you are five, six, seven, eight. Now x is negative, y is positive, which means you are in the sixth octet. Okay. Here, now this is positive means either you are in one, two, three, four, and x is positive, y is negative means means you are in the fourth octet. Is it fine? So I think octet also is clear to everybody. Nobody has any problem, any doubt with respect to octets. Great. So with this, we now move on to our distance between two points located in 3D space. Distance between two points located in 3D space. Can I move on to the next page? Any questions here? The distance formula, what we call it. Okay. So let us say there are two points A and B, which is x1, y1, z1. Okay. And B is x2, y2, z2. What is the distance between them? By the way, please note that all your writing devices, all the writing papers, et cetera, they are 2D figures. So making a 3D representation on 2D is slightly challenging. Okay. For example, it would not look good for me to say what is the distance between A and B, where A is x1, y1, z1, x2, y2, and B is x2, y2, z2. Okay. It doesn't look like a 3D figure. Okay. So please avoid using such kind of representations. Okay. This will only end up confusing you more. Right. So if I have to represent these two points, I would actually take the help of a 3D representation. How do I take the help of 3D representation? I'll show you. So this is not the right way to show two points. Of course, many times we tend to use this just to save time, but this is not how we should be using it. So what I normally do in order to show art in order to show the location of two points. Okay. What do I do is I use a cuboidal structure like this. I think this cube, this is longer. Yeah. So what do I do? I make a structure like this and x1, y1, z1. I place at one of the, you can say corners. Okay. x1, y1, z1. Okay. I'm assuming as of now x1, y1, z1, and x2, y2, z2. None of the two are common. I'm just assuming to be the most generic case. That means x1 is not equal to x2 or y1 is not equal to y2 or z1 is not equal to z2. Okay. So if none of the three coordinates are equal, I have to assume the other point to be on this corner. Okay. Now many people ask, sir, there are eight corners available. Right. So out of the remaining seven, why did you choose only this one? Okay. If I would have chosen this corner, let's say, correct? If I would have chosen this corner, tell me out of the x's, y's, and z's, which one would be equal to each other? Vishal, are you sure? Okay. To understand this, let me ask you a simple question. If I move in this direction, if I move in this direction, what will change? Assuming that this is oriented along x-axis, this is oriented along y-axis, this is oriented along z-axis. Okay. So this is our reference coordinate. If I move along this x-coordinate will change. Correct. So obviously this point will not have the same x-coordinate like this. So this would be x2. Agreed. This would change. What will not change? Will y change? How do you move any unit along this direction? That means inside the plane of the board. No, right? So this will still have y1. Correct. And this will be z2 because z-coordinate will change because you have to go up. Okay. But as I told you, I have assumed that none of the two coordinates here, x1 is not equal to x2, y1 is not equal to y2, z1 is not equal to z2. That means I have taken a most generic case. Okay. Even if they become equal, our formula will not get disturbed. Okay. So that is why I didn't choose this point. I chose such a point where not only the x-coordinate but y-coordinate and z-coordinate as well, they change. So B is the right position to talk about. Okay. Now everybody please pay attention here. What I need here is the distance between A and B. What I need here is the distance between A and B. Okay. So it appears to pass through this edge but it is actually not. Okay. How do I find this distance A, B? How do I find this distance A, B? That is my problem. How will I do that? So for that, I would ask you certain questions. Please respond to those questions. What is the coordinate of C? First of all, I want everybody to tell me. In fact, write it down on the chat box. What is the coordinates of C point? If this is A, this is B, what is C point? I would like everybody to write that down. Very good, Satyam. Very good, Arya. So C point will be x2, y1, z1. Correct. So as you can see here, nothing has changed. Just the x-coordinate has changed. Correct. Yes, Arun. Okay. D point, give me the coordinates. D point, give me the coordinates. Write it down on the chat box. Very good, Arya. Very good, Satyam. This is going to be x2, y2, z1. Okay. Now answer a simple question. What is the length? Let me just drag the figure. What is this length AC equal to? AC, that means this length of the cuboid, what is that length equal to? Write it down on the chat box. Okay. To be more precise, it's mod x1 minus x2. Correct. Am I right? What is this length CD equal to? Mod y1 minus y2. Excellent. What is the length BD equal to? Mod z1 minus z2. Correct. Very good. Okay. So now let me connect a line from A to D. Can I say this is 90 degree here? Okay. What will be AD length? What will be AD length? You'll say sir simple. We can follow Pythagoras theorem. AD square will be AC square plus CD square. Correct. Yes or no. So AD can be easily obtained from there. Right. Now focus on this triangle. ABD. And I claim that this is 90 degree. Am I right? So can I say AB square will be equal to AD square plus BD square? That is to say AB square is equal to AD square. AD square itself is AC square plus CD square. Right. So I am just putting this in place of AD square and BD square. I'll just write it separately. In short, what have I got? I've got AB square as AC square. AC square will be mod x1 minus x2 square. CD square is mod y1 minus y2 square. By the way, when you're squaring it, mod has no relevance. Okay. So you can skip writing mod. Okay. Unnecessarily. So what happens is that you get your AB length as under root of x1 minus x2 square, y1 minus y2 square, z1 minus z2 square. And to our happiness, this actually resembles the distance between two points in two dimensions. It's just that we have now an extra dimension to cater to. Right. We have z dimension also coming up along with x and y. Got it. Any questions, any concerns? Please note this down. Okay. So how are you finding this chapter so far? Easy. No problem. No custom. Okay. That is why I thought three hours is, you know, one finder will take it for three hours. Okay. Next Friday also will cover up one more topic called mathematical reasoning. So the Friday's maths class will keep. In fact, we have only three Fridays mass class. Okay. So today we are covering 3d. Our next Friday will cover mathematical reasoning. And maybe one more Friday, we will cover our topic, which is, okay, let's see whether we are able to do conic sections by then. So we'll see how much of conic section. Third Friday will keep for conic section, one of the conic sections, which is left questions. Let's take up questions. Let's start with very simple question. Very, very simple question. Just to get you started with the topic. What is the distance between, sorry, the distance between these two points is a root five, find the value of a, find the value of a, just to make you practice the formula. Easy. Okay. So this is just a touch and go question. Okay. Should I stop the poll now? Five, four, three, two, one, go. Okay. Nine of you have voted by the way. Why is such a list turned out in voting? Easy question. That's why. It's an easy question. We don't vote and all. Okay. So yes. So the answer would be under root of two minus four, the whole square, three minus three, the whole square, five minus one, the whole square. Correct. So this is nothing but two square, which is four and four square, which is 16. A root 20 is two root five. So A is two. Option number A is correct. Okay. Good. Let's move on. Next question is actually a locus based question. So you can get a locus based question on your distance formula concept. So what is the locus of a point which is equidistant from one, two, three and three to minus one. I will be relaunching the poll. Okay. Okay. Only four people have answered almost two and a half minutes gone. Okay. Maybe it is the first locus question. So everybody is trying to figure out how do I solve a three dimension locus question. 2D locus question. Most of you are aware how to solve. Okay. So let me do one thing. Let me solve this question that will give you an idea how to approach and I will follow it up with more locus questions. Okay. So should I solve the poll or anybody who wants to take a call. Most welcome to do it. Okay. Five. Four. Okay. Three. Two. One. Go. Okay. So. Confuses fonts. I can see a BCD. Almost. Almost equal words. Okay. So there's a confusion. Let's discuss the C. Okay. Let's say I draw these two points and be like this. Okay. And there is a point which is a moving point. Okay. Let me name it as H comma K comma. Let's say L. Okay. This point is moving in such a way that it's distance from these two points is equal. P is equal to PV. Just like we do locus questions in 2D. So I assume this point to be HKL. It's a moving point. But moving in such a way that it's distance from A and B as equal. Okay. So P is what just now I gave you a formula. Let me just name the code. Write down the coordinates here. P a means under root of, in fact, if P is equal to PV, we can say PA square will be equal to PV square. So we don't have to write under root and also it'll be H minus one whole square K minus two whole square L minus three whole square equal to H minus three whole square K minus. Two whole square and L plus one the whole square. Just cancel off the terms which are going to get cancelled like H square K square L square. They're all going to get cancelled. You'll be left with minus two H plus one minus four K plus four minus six L plus nine. This will give you six H plus nine minus four K plus four two L plus one. So nine nine gone one one gone. Okay. Four four also gone. And let's bring this H to this side. It'll be four H. Let's bring minus four K minus four K also gone. Let's bring this to the other side. This will be minus eight L. So I think everything is in taken care of. So you end up getting four H minus eight L equal to zero or H minus two L equal to zero. Now generalize. Generalize. How do you generalize putting H as X. K as Y. And L as Z. But depending upon whatever has survived, you will end up getting X minus two Z equal to zero. This is actually a plane, by the way. Later on, you will learn the equation of planes also. So plane basically has this kind of an equation. So mostly the plane equations look like this. Okay. It need not have all the X, Y, Z terms. Okay. So in this case, this is a plane. And this plane will be basically a plane like this. So every point on this plane, every point on this plane will be equidistant from A and B. I would like to share with you one important portal. Or you can say platform where you can draw 3D graphs also. That is called GeoGibra 3D. I think some of you already know about it. I think in the bridge course, also I had made a mention of it sometime. So let me show that to you. All you need to type is GeoGibra 3D calculator. Okay. So it shows you all the planes here, as you can see, and you can turn it around also. So let's say what was our point? One, two, three, right? Let me make one, two, three, right? Yeah, as you can see, it has shown here. This is one, two. One along x, two along y, and three along z. See this? Yeah. Okay. This is, what was the other point? I think other point was three, two minus one, right? Minus one. Okay. It's here. Three, two minus one. Okay. Now let me make the plane, which we got as our answer, x minus two z equal to zero. As you can see this plane. Okay. Let me give it a different color so that you can identify. Oh, you can see the color difference, right? One is green, another is green. So if you see this plane, it is equidistant. So this distance, this distance, the same as the distance. You see that? So this plane is your x minus two z equal to zero. Is it clear? Any questions? Any concerns? Let's take another one since I helped you in this question. I'll give you a few more questions to work on. Okay. Let's do this one. The equation of set of points p such that p a square plus p b square is equal to two ks. Where a and b are these two points respectively is does this question make sense? Or is something seems to be missing? Does it make sense? The equation of set of points p says that what sense it makes? It doesn't make sense, right? Okay. So I think some information is missing. Sorry. Yeah. Something is wrong. Sorry. I think some information is missing. Sorry. Yeah. Something is wrong. No. I also could not understand what does the question say? Okay. We'll take up another question on not to worry. We have other questions. Okay. Next is if the sum of the squares of the distance of this point from the two given points a zero zero minus a zero zero is two c square. Okay. Then which of the following is correct. Okay. Let's do this. I'll put the poll on. Poll is not letting you answer. But pull is on, but I did not get any response. Okay. Let me do one thing. Let me relaunch it. Now can you see the poll? Now I can see people answer. Okay. Sorry. Because of the technical glitch, maybe you are not able to answer. Done. This is a very easy question. Everybody should get this right. But why only four words? Please vote. Please vote. Five. Four. One. Or you need one minute for this. Okay. Let's discuss this. End of poll. By the way, seven of you have voted and all of you have voted for option. Option B only. Okay. Let's check. See what does it say? Let's say I call this point to be point B. Okay. And this point to be point A and this point to be point B. See what is given some of the squares of the distance is two C square. Right. So what is PA square? PA square is X minus a square. Y minus zero square. Z minus zero square. PB square will be what? X minus X minus minus means X plus a whole square. Y minus zero square. And Z minus zero square. This is equal to two C square. Okay. So if you see, if you combine these two terms, it will give you two X squared to a square. This two will give you two Y squared. This two will give you two Z squared. So you can cancel out. I think two factor from everywhere. So X square plus a square is equal to C square minus Y square minus X square, which clearly matches with option number B, which most of you have gone with. So B is the right option. Should not take much time. Simple. Is it fine? Any questions? Any questions? Any concerns? Okay. Should we take another one? Let's take this one. Find the locus of the points. Find the locus of the point. The sum of whose distance, the sum of whose distance, sum of whose distance is from four zero zero and minus four zero zero is equal to ten is equal to ten. Okay. Please simplify your answer before giving. Oh, so sorry Arya. Okay, Satyajit. Anybody else? Take your time. Take your time. Not sure. Dear all, just to summarize what all topics we have left to cover. We have to cover circles. We have to cover parabola, ellipse, hyperbola, mathematical reasoning, which we'll be covering in the next class. Apart from this, if I talk about the J syllabus, we have to cover binomial theorem. You have to cover trigonometric equations, properties of triangles. Okay. And pair of straight lines. But the last topics, which I have named since they're not coming for your immediate February exam, we can cover up when there is a break for your class 12th. That time also we can cover data. So my immediate priority would be covering up these five chapters, which is left. Most of these chapters are just hardly one or two classes. Okay. They are short only if I talk about school level topics. So I think I speculate by January and we should be able to easily finish it off. Okay. Should we discuss it? Done. Okay. So this is going to be hkl. Okay. The sum of its distance from A and B is given to you as a 10. That is to say under root of h minus four whole square k square l square plus under root of h plus four whole square k square plus l square is equal to 10. Okay. But now this is not a simplified version. We need to simplify it a little bit more. For that, what I will do is I will send the first term to the left hand side like this. Okay. And then I'll square both the six. So this will give you if I'm not mistaken, h square eight h 16. This will give you 100. At square minus 20. Okay. So Tim, so Tim wants to change his answer. Okay. Let's simplify. 16 gone. 16 gone. This gone. This gone. Square gone. 16 gone. This gone. Correct. Send this eight h to the other side. So eight h will come as 16 h. Okay. It's minus 20. You may easily drop a factor of four throughout. Now, just stop the positions like this. So five is five under root of this is equal to 25 minus four h. Now square both sides. So if I'm not mistaken, this will give you 25 times h square minus eight h plus 16. Is it fine? All right. So let's now start clubbing the term. So you'll end up getting nine h square. Correct me if I'm wrong. You'll end up getting 25 k square. We'll end up getting 25 L square. Okay. And I think minus 200 h minus 200 h that will also get cancelled off. So you will end up getting 25 into 16. 25 into 16 is 400. So 400 and 625 will adjust themselves as 225. Okay. So now here you can generalize it. So when generalizing, you replace h with x k with y L with z. This is the answer. Okay. By the way, I don't think so. Anybody got this right. None of you got this right. I was expecting somebody to get this right. Is it fine? Any questions, any concerns? Okay. So we'll be starting with the section formula after the break. So we'll take a small 10 minutes break, not much because the classes are shorter duration. So 602 right now we'll meet at 6 12 p.m. Okay. Exactly after 10 minutes, a short break. So eat and drink something. We'll see you in 10 minutes time. So the next concept that we are going to talk about is your section formula. So how does the section formula work in 3d geometry? Like what we had learned in our 2d geometry, there was some section formula. So how is it in 3d geometry is what we are going to study in this particular topic. Okay. So let us say there is a joint of two points. Let me call it as point A point B. Okay. Let's say point A, let's say has coordinates X1, Y1, Z1 and point B, let's say has coordinate X2, Y2, Z2. Okay. Now if there is a point C, if there is a point C, let's say here, which divides the joint of A and B in the ratio of M is to N. So let's say this ratio is M is to N. Then what is this coordinate of C or what is your X3, Y3 and Z3 in terms of X1, Y1, Z1, X2, Y2, Z2 and your MNN. In other words, we want to find out the coordinates of C in terms of the coordinates of A and B and of course MNN. How do I find this out? Okay. So for this, I will be deriving the result for all of you. Please listen to this result. Okay. A lot of things you will be learning in this. Okay. So everybody, please pay attention to the derivation part of it. Let us say this is our coordinate axis. Okay. Let's say this is our X axis, Y axis, Z axis. Okay. And this is a line in the 3D space. So you can assume that this is your, you know, you can see my camera here. This is your line in 3D space. Okay. Like this. Okay. Now, what I'm doing is I'm dropping perpendicular form A, C and B on the XY plane. Okay. So let's say this blue line is your perpendicular. Is it blue or red? Sorry. Blue. Okay. So what I have done is I have dropped a perpendicular or you can, you can call this line to be a projection. So A dash, B dash and C dash, you can call them to be the projection of ABC on the XY plane. So think as if this is your line. Okay. Look at the camera. This is your XY plane. And this black pen over here is the projection of this line. So that means that this tip, which I'm showing with my little finger. If you drop perpendicular, it will hit on the tip of this particular black pen. Okay. And this tip, which I'm showing with my index finger. If I drop perpendicular, it will hit here in the back of this pen. Okay. Are you able to imagine it? Everybody is able to imagine this? No. Let me ask you a few questions here. I'm sure you will be able to answer them. What is the coordinate of A dash? Who will tell me? What is the coordinate of C dash? And what is the coordinate of B dash? Who will tell me A dash coordinates? Now that you're expert in finding that, imagining the 3D space. Tell me. Very good. A dash is X1, Y1, 0. Very good. C dash, X3, Y3, 0 and B dash, X2, Y2, 0. Please don't forget to write the Z coordinate. That will be 0. Fine. Okay. Now everybody, please pay attention. Here is a very important part. Now what am I doing? I am just lifting this image. So basically this was a situation where there was a line and this black pen was the image of that line. Correct. Now what I'm doing, what I'm doing, I'm taking this black pen and I am lifting it like this to make it a X kind of a structure over it. So this image which was down on the XY plane. I hope you can see it. If you want, I can just unshare my screen. All of you are able to see me. Is the view bigger? Is the view bigger now? So this was your image. The black pen was your image. Okay. And this blue pen was your actual line. I hope you can see it. Now what I'm doing this, I'm taking this image or the projection on the XY plane. And I'm lifting it up to make the two fingers meet like this here. Are you getting the point? Okay. So let me show that on the diagram. So I'm lifting this and making it come like this. So in short, this is parallel to this line. So this C dash and C, I'm making them meet. Okay. By lifting this blue line. Okay. I'm making C and C dash meet with each other. So this kind of an X structure is created. Let me call this point now as a double dash and let me call this point as B double dash. Okay. Who will tell me a double dash coordinates and who will tell me B double dash coordinates? Note that C and C dash have actually met each other. So they are at the same point. So what is A double dash coordinate and what is B double dash coordinate? Please write it down on the chat box. What is the coordinate of A double dash? No, Aria. No, no, no. So A double dash and A will be at the same position though, but they're not in the same positions. See again, try to imagine this. Okay. This is your image and this was your original line. Okay. You lifted it up like this, like this. No, Aria. No. That is still. Satyam has given the right answer. It will be X1, Y1, Z3 because you have lifted, you have lifted this image. See again, I'm showing you. You have lifted this image by Z3 up. Every point has gone Z3, Z3 up. So A dash has gone to A double dash and it will become X1, Y1, Z3. Are you getting my point? Similarly, B double dash will become X2, Y2, Z3. Is it fine? Any questions? Any concerns? Any questions related to A double dash and B double dash coordinates? Please let me know. Any doubt? How did I get X1, Y1, Z3 and X2, Y2, Z3? If you have any doubt, please ask me again. Okay. Now here I'm going to create a triangle. In fact, two triangles I'm going to create. One triangle that I've created is A double dash C A and the other triangle that I've created is B double dash C B. Now, do you all appreciate the fact that they will be similar triangles? Yes or no? Agreed? Everybody agrees? A double dash C A that is this triangle will be similar to this triangle. I don't want to dirty the figures. I'll just remove it. Agreed? Okay. Now if that is the case, can I say AC by BC is A double dash A by B double dash? Would you agree with this? So since this is similar to this, it implies this will be true. AC is M, BC is N. Correct? As per our this thing, this by this is M is to N. I've already written it on the chat on the screen here. Okay. And what is A double dash A? A double dash A, the distance is the difference in their Z coordinates. And similarly, this will be Z2 minus Z3. Am I correct? Any questions? Okay. So from here, I will make Z3 the subject of the formula. How? Very simple. Bring this NZ1 to the left side. So MZ2 plus NZ1. Bring this to the other side. It will become M plus NZ3. So there you go. There you go. You get Z3 as MZ2 plus NZ1 by M plus N. Okay. Do you all agree with this simplification? If you have any doubt anywhere, do let me know. Okay. Now I have got this Z3 in terms of Z1, Z2, MNN. By taking the projection of this line or by dropping perpendicular from these points on to the XY plane. Had I done the same activity by drawing perpendicular on YZ plane, I would have actually got X3 in terms of X1, X2, and MNN. So that I will simply write over here. Similarly, similarly, by dropping perpendicular to YZ plane, you may prove that or it can be proved that or it can be shown that X3 is MX2 NX1 by M plus N. I'm not deriving it again. The process is exactly the same. It's just that now you have to drop perpendicular on the YZ plane and do it. Okay. And similarly, by dropping perpendicular to to XZ plane, it can be shown that it can be shown that your Y3 is MY2 plus NY1 by M plus N. Okay. So let's not repeat the same procedure again and again. Since from your Z idea, you will, from your expression of Z3, you would get an idea. How would you get X3 and Y3 in a similar way? Right. So now what I have done, I have actually found out the X3, Y3, Z3 means the coordinate of C. So that is your C coordinate has now been figured out. And that is nothing but MX2 NX1 by M plus N, comma MY2 NY1 M plus N, MZ2 NZ1 by M plus N. Now let's say, sir, this is also very similar to our section formula in 2D. In 2D we only had these two. Okay. Now a third one has been added up. Okay. So it makes you remember all these formulas very easily because there is a close connection with your 2D geometry as well. Okay. So please note this down. Now remember here, this same formula will work for external division also. So if you have something like A, B, C here, this will be an internal division. We already know that from our 2D understanding, internal division. Okay. And if you have something like this, A, B here, then C here, then this is called external division. So C divides joint of A and B externally. So in this case, this is your M. This is your N. Okay. So M by N is considered to be positive. In this case, this is your M. This is your N. And in this case, M by N is considered to be negative. Remember the concept of positive negative. Okay. So same formula will be applicable even for external division is just that now the ratio M by N will be actually a negative quantity. Okay. Is it fine? Now you may write this like this also, you can divide throughout with an N and write it like this. So this can M by N can be written like a K also. So you can write this as KX2 by X1 by K plus 1. KY2 plus Y1 by K plus 1. And KZ2 plus Z1 by K plus 1. Remember this K will be negative. Okay. So K will be positive for internal division and K will be negative for external division. I've already told this in your 2D. So I don't want to repeat this again. Same procedure is applied for 3D as well. No difference. No difference. Is it fine? Any questions? Any concerns? Please do let me know. No doubt. Do let me know. Everybody is done. Should we take questions now? Okay. Great. Let's start taking questions. Let's begin with this question. The ratio in which the ratio in which the join of these two points is divided by XY plane is which of the following options? So I put the poll on. Okay. Good. Just two people have responded so far. Please note that the ratio that they mentioned is always see as many people will say, sir, these two look the same to me. They're actually not the same. Let me tell you why. Let's say this is your A point. Okay. I'm just giving am I mean, don't consider that I'm trying to claim that this is the right answer. But if at all there's a confusion like this, one is to three means one is towards this side. So this is one and this is three like that. Okay. So one is that one is towards the first point and three is towards the second point. Okay. If you can visualize and solve it, that's good enough. Okay. Should we stop the poll now? Five of you have responded five, four, three, two, one. Go. Okay. Not many people have responded, but those who have responded, they have all given only one response and that is option B. Okay. So most probably this should be the right answer. Let's check. So since I don't know what is this ratio, I will take it to be K is to one. Okay. K is to one. This becomes smiley. Okay. Right. So let me just solve it in a proper way. So as per this ratio, your coordinates of C will be now all of you, please pay attention. Okay. Many of you would be wasting time writing all the coordinates. Okay. That's what her shit is pointing at. But if it is divided by the XY plane, this is your XY plane. Let's say. Okay. On your XY plane, Z coordinate is zero. So C must have Z coordinate as zero. Okay. So Z coordinate of C must be zero. So only write only focus on writing the Z coordinate. Why do we waste? Why should we waste time writing, you know, X and Y? Let it be anything. Z coordinate. I'll focus only. So it's K into minus one. One into three by K plus one. And this should be equal to zero. And if this is equal to zero, that means minus K plus three by K plus one is equal to zero, which means K should be equal to three. So this ratio is three is to one, not one is to three. Okay. So option number B will be the right option. Is this fine? Any questions? Any questions? Any concerns? All right. Let's take this question. A, B and C are three points forming a triangle. And AD, the bisector of angle BAC meets BC in D. Find the coordinates of D. All is on. There's two people have responded so far. Okay. Anybody wants to answer now because I think end of time has been given. Can you discuss? Okay. Great. So I'll stop the poll now. Anybody wants to vote, please do so. Okay. I think only five of you have chosen to respond. And our five of you, two of you say A, two of you say B and one of you say C. Let's check this out. Let's say I draw this figure. ABC like this. This is your triangle ABC. So A is three, two, zero. B is five, three, two. And C is minus nine, six, minus three. Now this angle BAC is bisected by a line AD. Now AD hits BC. Okay. Add D of course. What is the coordinate of D? That is what we are looking for. How would you solve this question? Yes. Right. By angle bisector theorem. Exactly. So by angle bisector theorem, we have already studied this theorem during our childhood days. Correct. Angle bisector theorem. It says that if there is a bisector internal bisector of an angle, it meets the opposite side in such a way that BD is to DC is AB is to AC. Can you say distance of BD is equal to DC? No area. Okay. Let me show you our extreme case. Imagine a triangle like this. Okay. You ask yourself if I bisect this angle, let's say. Okay. Let's say this angle is this angle. Does it mean BD and DC are equal? No, right? So you can't say BD is equal to DC unless until it's an isosceles triangle. Okay. Is it fine? No. For that, I need to know what is AB and AC. So what is AB? Let's figure it out. So AB will be under root of five minus three whole square, three minus one whole square, two minus zero whole square, which is going to be three. Correct. And what is AC? Let's also figure that out. So AC will be nine plus three, the whole square, which is 12 square, so two minus six, the whole square, which is four square and three minus zero, the whole square, which is three square, which happens to be 13. So this ratio is three is to 13. That means this is three. This is 13 in terms of ratio. Now, once you figured out the ratio, which is three is to 13, you can use your section formula, right? And as for the section formula, the answer to coordinates of D will be, let me write it down over here. So D coordinate will be three into minus nine, 13 into five by three plus 13. Correct. Then three into six, 13 into three by three plus 13. Okay. Three into minus three, 13 into two by three plus 13. Okay. So this will result into minus 27 plus 65 minus 27 plus 65 is 38. Okay. So 38 by 16. This is going to be 39 plus 1857 by 16. And this is going to be 26 minus nine. 26 minus nine is 17 by 16. Does this match with any option? 38 by 16 is also 19 by eight. Oh yes. So they have simplified it. So this is 19 by eight, 57 by 16, 17 by 16. So option number B is correct. Any question, any concerns here, do let me know. Is it fine? Now what is the easier method that Harshita you were talking about? Same method. Okay. Oh, and what was it? And what was the wrong method that you were applying? Let me know that as well. If at all you want to share. Sir, I started solving equations. Sir, I took it as XYZ and then I started solving it. No, no, no, no, no, no, that's the biggest mistake. And then I realized it's wrong. Using angle bicycle theorem itself. I forgot this K that you can take the ratio and then I realize you can do that. Yes. So please realize that you have not learned the equation of 3D lines. Okay. That's a different way all together. It's not the same as what you have learned in your 2D. It's all similar, but not exactly the same. Let's do this one. This one. The ratio in which why is it pain divides the line segment. This is to is to M. Find the value of M. I mean, almost similar type. Should we put the poll on in case you're done. Right area. Yes. Done. Okay. Should we start the countdown five. Four. Three. Two. One. Okay. Seven of you have responded and all of you responded with option number B, which is three. Let's check. So it's very simple and why is it pain if a point lies. It's X coordinate will be zero. All right. So let us say this ratio is two is to M. Okay. This ratio is two is to M. So this coordinate only X coordinate. If you write it will be two into three. M into minus two by two plus M. Okay. Rest of the coordinates. I need to not worry about them. Okay. All I need to worry about is that putting this as zero, which means six is equal to two M. M has to be three simple. Any questions very similar to one of the questions we did. Next question, which I'm giving you, I'll be writing it down for you. Find the ratio in which. The sphere. The sphere. X square plus Y square plus that square is equal to 504. Divides the. Divides the. Join off. These two points. Okay. Find the ratio in which the sphere. I don't worry too much about this fear. You will be singing yourself has not got sphere and all he is giving this fear. It has nothing to do with your knowledge or sphere. Okay. Think that as if there's a curve, which is dividing the joint off. These two points. Okay. What is the ratio in which it is dividing. Okay. Okay. See all of you please. There is a connect of two points. Let me write them down. And this connect is basically. Okay. Divided by a sphere. Okay. Now all of you please pay attention. The sphere. I mean, I'm not, I don't know the exact position of the sphere right now, but the sphere can basically. Cut the point. Cut this line. Maximum at two point. So it could be like this. Let me make a different color sphere. Okay. Now many people will say it is only cutting at one point. Actually it is cutting at two point here. One point is this. And the other point is a case of external division. So this is internally. Cutting it. And this is externally cutting it. Okay. So it could be like this. Let me make a different color sphere. It could be like this. Okay. Now many people will say it is only cutting at one point. Actually it is cutting at two point here. One point is this. And this is externally cutting it. Correct. Or it could be like this also. I mean, I don't know what is the position right now. I'm just speculating that these are the possibilities. And not only that, it could be like this also. How many people will say, sir, here it is not even cutting it. It is cutting it when both are external divisions. Okay. So this point, let's say I call it as P dash. So P dash and Q dash both are external divisions. Okay. Now only calculations will tell us what is the exact scenario, but you will always get two answers for this question. Either both the answers will be positive or both the answers will be negative or one will be positive, one will be negative, but two answers you will get for the damage for that ratio. Right now I can see only one person giving me two responses plus minus. And all of you are giving me one one answer each. So people who are giving me one one answer each. Definitely your answer is not going to be right. Check. So basically there could be three possible ways in which the seer can can divide the joint of A and B. It could be like this, the case number one, where is one is internal or one is external, or it could be like case number two where both are internal. So this is internal internal. And it could be either case number three where both are external. Right. That only calculation will tell me what is going to happen. So when I solve this question, I will normally assume that let the, let the point where, so I'll just re sketch this whole scenario once again. Okay. Let's say the point where the sphere is cutting the joint of A and B. That is in the ratio of case to one. That is in the ratio case to one. That's it. This is your C point. So coordinate of C will be what? What it is? He will be what? 27 K plus 12 by K plus one. Okay. Minus nine K minus four by K plus one. And 18 K plus eight by K plus one. Okay. Now this point must satisfy. The equation of the sphere. Am I right? Yes or no. This point must satisfy the equation of this year. So now when I put it in the equation of a sphere, that is X square. What was the equation of this year? X square plus Y square plus X square equal to something. No equal to 500 and four. So let's use that X square. Y square. By the way, there's no point writing a minus sign within a square. So we can just make a plus X square, Y square Z square is equal to 504. Okay. Okay. Now this is going to give me a huge expression. All right. So I have a trick here. I can see a lot of resemblance between nine, four, 18, eight, 27 and 12. So what I will do is I'll say let nine K plus four by K plus one be lambda. Okay. When this is lambda, then automatically can I say 27 K plus 12 by K plus one. It will be three lambda. Am I right? Yes or no. And automatically we can say that 18 K plus eight by K plus one will be two lambda. Do you agree with me? So that makes our life super easy. So in this equation, when you put you get three lambda square, lambda square to lambda square is equal to 504. Okay. That gives us 14 lambda square is 504. So lambda square is 36. Correct. So lambda is plus minus six. Now you don't have to waste your time doing the simplification by taking the K plus one square to the other side and expanding it. That is going to waste a lot of your time. Instead, this is going to give us the answer in a much time efficient way. Everybody's happy so far. Now once you've got lambda values, you're not comparing nine K plus four by K plus one with your lambda. So this could be six or this could be minus six or two possibilities are there. We'll compare both the possibilities and see K value. What does it come out? So from here, I will get nine K plus four is six K plus six, which means three K is two. So K is two by three. So one ratio is two is two three. So from here, I'll get nine K plus four is minus six K minus six. So 15 K is minus 10. So K is minus two by three. So the ratio is, so this is two is to three internally and this is two is to three externally or external division. So now tell me which of the three cases actually happening in this first case, second case or the third case, first case, right? So this ratio, this is two is to three. Okay. And this is two is to three. Getting the point. So this is external. This is internal. Clear. So equations will automatically tell you the exact picture. Any question, any concerns, anything that you would like to copy from here, please do so. Okay. Fine. So let's extend this concept to find the centroid of a 3D triangle. Okay. All of us know what is the centroid? Centroid is basically nothing, but it's the meeting point of the medians. Okay. So how do you find out the coordinates of the centroid for a triangle? Whose coordinates are in 3D space? Okay. By the way, many people ask me, is it a 3D triangle? Can a triangle be a 3D figure? Tell me. Anybody? Can a triangle be a 3D figure? Just like sphere is. Can a triangle be 3D figure? Yes or no? No, it cannot be. It's a 2D shape. Even if it's three coordinates, even if it's three coordinates are having XYZ, it doesn't mean it's a 3D figure. Don't get confused here. Because the lamina of the triangles can be in single plane only. Are you getting my point? So never say 3D triangle. Triangle is 2D only, 2D figure. Are you getting my point? So how do I get the coordinates of the centroid? How do I get the coordinates of the centroid? So please note that the concept of the centroid G dividing the joint of let me call it as PQR. So the centroid divides the medians in the ratio 2 is to 1. That will still be valid for this. 2 towards the vertex side. Can you tell me what would be the coordinates of the centroid? Please everybody find this out. Now you know your section formula very well. Done. So simple. This is the midpoint. So this midpoint will have coordinates of X2 plus X3 by 2, Y2 plus Y3 by 2. It's a midpoint. And now see G divides AQ in the ratio 2 is to 1. So for G the coordinate will be, let me write it again. For G the coordinate will be 2 into X2 plus X3 by 2 plus 1 into X1 by 2 plus 1. And similarly 2 into Y2 plus Y3 by 2 plus 1 into Y1 1 into Y1 by 2 plus 1 and 2 into Z2 plus Z3 by 2 into 1 into Z1 plus 1 into Z1 by 2 plus 1. Which yes comes out to be the average of the X, Y and Z coordinates. Exactly. So this is the, now again this formula is very, very similar to what we have in 2D. It's just that now we add a third dimension to it, which is of Z. Okay. So this is the coordinate of the centroid. Please note it down and remember it because all of questions will be directly asked. Is it fine? Any questions? Okay. Should we take a question now? All right. Let's take a question. Very simple question to begin with. Origin is the centroid of ABC having vertices at the given coordinates. Then which of the following option is correct? Simple. Expecting the answer within 30 seconds, actually to be very frank. Very good. Should we discuss it now? Should I end the polls in the count of five? Okay. Five, four, three, two, one, go. Okay. People have responded with CND as well. I don't know why, but let's discuss it. Most of you have said option number A. So it's very clear that the centroid coordinate will be A minus two plus four by three, comma, one plus B plus seven by three, comma, three minus five plus C by three. Okay. And each one of them is equal to 0, 0, 0. Which means A plus two is zero, which means A is negative two, which means B plus eight will be zero. That means B is negative eight. That means C minus two should be zero. That means C should be two. And out of this only A is correct, B, C, B are not correct. Is it fine? Any questions? Take this question. Let's say there is a triangle. There is a triangle A, B, C. The midpoints of A, B is one, two, three. Midpoints of B, C is three minus two, one and midpoint of C. A, C is zero, four, zero. Find centroid coordinates of triangle A, B, C. Find the coordinates of the centroid of triangle A, B, C. Okay, Arvind. Okay, Arshad. Okay. Let's discuss this. I think many of you are taking a longer time to solve this, which is ideally was not expected. This is actually a super easy question. See, let me ask you this question. Let me call this midpoint as PQ. Okay, and let's say I call this point as R. Do you all agree that if I draw a line from A to R, R being a midpoint, can I say the same line will also be, the same line will also be dividing PQ in the ratio one is to one or it will be passing through the midpoint of PQ. Do you all agree with that or not? If no, then I can give you the explanation. If yes, then I can move on. Okay, if no, then I can just tell you that APQ and ABC will be similar triangles, Thales theorem. I'm sure most of you would have done Thales theorem in your class 10 or basic proportionality theorem. So if you connect the midpoints, this will be parallel to this and hence this angle will be equal to this angle and this angle will be equal to this angle. Correct? Yes or no? Yes? Correct? Same will go with this as well. In short, this length will also be half of this length and this length will also be half of this length. In short, basically PQ is half of BC also. Agreed? In the same way, can I say, had I connected this and I would have drawn a line connecting B to Q, then this will also be bisecting this. Correct? And same will be working fine even if I had connected Q to R and I would have drawn a line from C to P, then this line would have also bisected this fellow. So what do you observe here? We observe that the centroid of ABC is also the centroid of PQR. Okay? This also happens to be centroid of PQR. Agreed or not? If I have to find the centroid of ABC, why not just find the centroid of PQR, which is nothing but 1 plus 3 plus 0 by 3, 2 minus 2 plus 4 by 3, and 3 plus 1 plus 0 by 3, which is just 4 by 3, 4 by 3, 4 by 3. Does it take that much of the time? How many of you actually went to find the coordinates of ABC and then find out the centroid? Aiyo, Satyam! Anyways, is it fine? Any questions, any concerns? Alright. Now just out of curiosity, I would like you to, I'd like you to solve a question. Okay. Maybe you would have done that in physics to a certain extent. Yes. So that 2D formula will still be valid for this. I don't know why did you change that answer. Okay. Let's say, let's say I have a triangle. I have a triangle. Let me give you some coordinates. Yeah. Let's take this coordinate. 5, 4, 6 for A, 1 minus 1, 3 for B, and 4, 3, 2 for C. Okay. Find the area of the triangle. Find the area of this triangle. Something in vectors that you have done will help you to get this answer. I just want to see how many of you remember that, if at all you have done that. Have you all done cross-product in vectors, right? If you have done cross-product, then you should be able to get this answer. If you have done cross-product, then you should be able to solve this question. What is the area of this triangle whose coordinates are in 3D space? Okay. Anybody? So first, let us go to the concept of vectors. I like to ask you a simple question. If this is vector A and this is vector B. What does mod of the cross-product of these two vectors give you? What answer does it give you? Many of you still say, sir, mod A, mod B. Correct? But have you ever figured out the geometrical meaning of this? Right. Geometrical meaning of this is nothing, but it's the area of this parallelogram. Right? So this area is given by this expression. Everybody knows it? All of you are aware? Okay. Now, let me do one thing. Let me just cut this parallelogram like this and make a triangle out of it. So can I say the area of this triangle, area of the triangle, will be half of mod of A cross B? Correct? This actually will help you to solve this question. So if you know any two vectors, let's say I find this vector and this vector. That is BC vector and BA vector. And take the cross product of it, take the modulus and half my answer, I should get the area of the triangle ABC. Do you all agree with me on this? Yes or no? Anybody having any doubt in this? Okay. No doubt. Okay. So tell me BC vector. Anybody can tell me BC vector. Quick, quick, quick. Write down BC vector, BC vector, BC vector, BC vector, BC vector, BC vector. 3i cap plus 4j cap minus K cap. No, that is not BC vector. You are writing the position vector of that point. This, this what you have written. This, if you convert it to a vector, it becomes the position vector of that point. This is also position vector of C. So position vector. So let me write it down like this. BC vector is position vector of C minus position vector of B. Position vector is not that clear. I have to tell the rest there. The students are not clear about position vectors. Okay. Anyways, be a vector. Tell me. Tell me. Tell me be a vector. I should show you this point. 546. Yeah. Write down be a vector quickly. Character event 4i plus 5j plus 3k. Okay. Now, once you know two vectors, how do you find out their cross product? I hope you have been taught the determinant vector to find the cross product. Everybody knows. Okay. Sure. 100%. Okay. So let's expand it. So still plus 517 minus J9 plus 413 and plus K15 minus 16, which is one. So your answer will be half modulus of this. B is a position vector minus B position vector. So four, five and three. It's always destination minus source. So where the vector is going, that position vector minus where it is coming from. So your answer will be half under root of what is the magnitude of this vector? Yes. Got it. I mean, you can, you know, calculate it by the way, 17 square. Let me just do it. 389, 289 and then 169 plus one. So it's under root of 459 square units. Ugly figures are coming because I just made up the question. But is the idea clear how to find out? Okay. So any question related to finding area of a triangle whose coordinates are in three dimension. Don't get scared. You can use your cross product fundamentals to solve it. Okay. So with this, we close this chapter and we are done with one chapter, one small, simple chapter for 11th. Okay. It's already done. See you tomorrow. With conic sections. So we'll start with circles tomorrow. Okay. I'll send you an official reminder. 4 to 7, 30 class. Bye bye. Take care. Good night. Bye everybody. Thank you.