 Introduction to 3d geometry, yes that name of the chapter is not 3d geometry it's actually introduction to 3d geometry for the simple reason that you are actually 3d geometry is going to come in class 12th okay three-dimensional geometry this is just an introduction there's just a know-how of you know what is a 3d space and what what do the coordinates in a 3d space mean okay what is the section formula what is the distance formula etc so this chapter is not very long so I'm sure I can complete this in half an hour or 40 minutes so this chapter has primarily got only two concepts into it one is the distance formula of course there is a basic know-how and after that the main concept is distance formula which is just a one or two minute you know discussion and then there is a section formula okay with a section formula so under section formula we are also going to touch upon the concept of centroid of a triangle okay so that is also sometimes asked the main part of 3d geometry lies in class 12th where you will be studying about the concept of direction cosines ratios equations of lines equations of planes and various interaction of lines and things and let me tell you that 3d geometry will be along with vectors okay so there will be a touch of vectors in 3d geometry in class 12 right now it is just not even 5% of what you're supposed to know actually so before I start anything please recall what we had done in our 2d geometry in 2d geometry we basically had two reference lines which we used to call it as the x and the y axis correct and any coordinate had got a and b as the coordinates or any point would get would have to a two parameters a and b where a is called the abscissa b is called the ordinate the mod of the abscissa actually represents how far is the point from the y axis okay and mod of basically and we are directed distances to be you know short and simple okay so in case of 3d geometry we would require we would require three axes right so these three axes are basically oriented in a special manner which we call as the right-handed coordinate system yes so right now on your screen I have drawn a perspective view of x y and z now this angle is 90 okay this angle is 90 and this angle is 90 I hope you can see from the figure let me draw it like this okay but these 90 degree position of these three lines are not arbitrary they're not like chalo man kar tha to koi bhi three perpendicular lines you know you can draw no not like that it's not based on your whims and fancies okay these three coordinate axes are basically oriented in such a way that they form a right-handed coordinate system a right-handed coordinate system what is a right-handed coordinate system right-handed coordinate system is basically a system where if you let's say stretch your right hand in such a way that the edge of your hand is kept on the x-axis okay so I hope you can see the camera so basically the edge of the hand should be kept on the on the x-axis and then curl your finger curl your finger naturally towards your y-axis so curl this finger naturally or curl these four fingers naturally towards your y-axis in that case your thumb direction will be along the positive z-axis so what you see over here is a positive x positive y and positive z okay so we cannot draw our x y z-axis in any random fashion that we want to now there may be cases where the question setter will say you are following a left-handed coordinate system so the question will start with the disclaimer that assume that you are following a left-handed coordinate system in that case basically it is going to be the other way around that means if you curl your finger naturally or you can say you can take your left hand left hand and then curl your finger towards the y-axis then your thumb direction will be towards the z okay that will become a left-handed coordinate system but if nothing is mentioned you will follow a right-handed coordinate system please be aware of that okay now these right-handed coordinate system it divides our 3d space it divides our 3d space into octants now in in front of you I have basically shown a figure okay so imagine imagine as if you have a cubicle kind of a structure like this I hope you have seen cubicles in offices right I'm sure if you have visited an office in your school also you would see cubicles like this where people are sitting so here this is your x-axis this is your x-axis this is your y-axis direction this is your z-axis direction now this entire 3d coordinate system divides the space into eight zones which we call as octants which we call as octants it's just a second lot of noise coming from outside yeah so now octants just like we have coordinates in 2d system we have octants in 3d system now what are the name of the octants all of you please note down this is called the first octants as you can see this cubicle I'm trying to shade it this cubicle is your first octant okay in first octant everything is positive as you can see the sign over here x is positive y is positive now before I go to the signs let me just let's say this is our right handed coordinate system if you have any point in space let's say I call it as a comma b comma c okay in 3d coordinate system you need three coordinates a b and c a basically represents a basically represents the directed distance distance from yz plane I hope you can see this plane this plane is your yz plane I hope you are able to imagine it okay so a represents the directed distance from the yz plane now what is this direction if this is in along the positive x-axis it's a positive value if it is toward the negative x value it is a negative value so from the yz plane a represents the directed distance similarly b represents the directed distance from xz plane so xz plane is the one which is in front right now the screen of your laptop or your desktop whichever you are using that is showing your xz plane okay and c represents the directed distance from from xy plane xy plane is the bottom plane okay so c represents the directed distance from the xy plane so here in 3d geometry you have x so you have a b and c as three coordinates and if you are lying in the first octant then all of these will be positive okay all of them will be positive as I have shown it over here in the diagram also okay this is called the second octant in second octant x is negative y and z both are positive okay this is your third octant in third octant only z is positive both x and y become negative okay this is your fourth octant fourth octant x is positive y is negative z is positive the octant which is right under the first one is the fifth octant the octant which is right under the second one is the sixth octant the octant which is right under the third is the seventh octant by the way six I will not be able to show because it is hidden from the view okay and this octant which is below your fourth one is your eighth octant I hope you can note down the octant's position so in the exam they may ask you what is the sign of x y z in let's say third octant in third octant x and y both are positive negative only z is positive what is the sign of x y z in the seventh octant seventh octant everything is negative okay so such kind of questions can be asked to you so I hope there is no issue regarding the positioning of these octants and the sign of your ABC in these octants any questions okay now the next concept that we are going to talk about as I told you I will directly come to the point distance formula this is the very first concept that we normally study when we are talking about coordinate system so in 2d also I'm sure in 9th or 10th you would have studied about distance formula as the first thing so if I ask you what is the distance between two points x 1 y 1 z 1 and x 2 y 2 z 2 now it's a big challenge for teachers to actually represent a 3d point on a 2d space so normally what do I do I hate representing a point like this in 2d space so what I do I normally fix my coordinate axes like this I hope you are you'll be able to understand so this is your x this is your y this is your z by the way please imagine this is a perspective view that means this is right angle this is right angle and this is right angle okay this is a perspective view okay then what do I do let's say I have a point okay then I make a I make a you know cubicle box like this so that you understand that I'm referring to a 3d space okay yeah let's say I make a structure like this okay so let's say this is your point x 1 y 1 z 1 now I have oriented this cube in such a way that this is parallel to the z axis this is parallel to the x axis and this is parallel to the this is parallel to the y axis okay so I have oriented this box in such a way that this box side is parallel to x this box side is parallel to z and this box side is parallel to y okay and let's say this is point a and this is point b so let me call this point as x 2 y 2 z 2 now I want to find out a b distance now all of you try to imagine this in the room that you are sitting in all of you you're sitting in a room I'm sure nobody's sitting in a lawn or something I cannot deny it by the way some of you maybe so imagine that on the floors one of the corners of the floor is your x 1 y 1 z 1 point okay and the opposite diagonal in the ceiling is b point x 2 y 2 z 2 and you want to find out what is this distance you want to find out this distance a b okay now how will you find it out very simple first of all what is the coordinate of this point c I'm sure everybody will agree that c will have the same x and the y coordinate as b but same z coordinate as a because it is on the same floor see on the floor of your room all the points will have the same z coordinate isn't it unless until your floor itself is you know inclined okay so imagine that your room is says that the edges of your walls are behaving as your x y and z axis of course forming a right-handed coordinate system okay so this point c will have the same x and the y coordinate as this but same z coordinate as this okay so obviously this distance bc if I'm not mistaken will be z 2 minus z 1 to be more safe put a mod so that you know you assure that it is a positive distance okay what about this distance ac so when you talk about ac it is basically as if you have two points which have the same z coordinate they are only different differing x and y coordinates so this point is this line will have a length of x 1 minus x 2 whole square y 1 minus y 2 whole square you will be basically using a 2d formula over here because this length is nothing but as if you are trying to connect x 1 y 1 with x 2 y 2 because z is not varying at all okay now look at the triangle a b c this is a right angle triangle okay this is a right angle triangle a right angle at c okay so this angle here I hope you're able to understand this angle is right angle right so in a right angle triangle we can apply our Pythagoras theorem that is you can say ab square is ac square plus bc square ab square is let's say not known to us ac square the square of this will be nothing but x 1 minus x 2 whole square y 1 minus y 2 whole square bc square square of this you can actually write it as z 2 minus z 1 whole square or z 1 minus z 2 whole square doesn't make much of a difference and there you go this is your formula for the distance between two points in 3d geometry now this is very easy to remember because it is just an extension of the same formula that you have learnt in 2d is just that one more dimension is added to this one more dimension is added to this is this fine any questions okay we'll just take a very simple question on this not a big question question is find the locus of a point the sum of distances from 1 0 0 and minus 1 0 0 is equal to 10 so find the locus so here also locus questions are not leaving you so find the locus of a point the sum of distances from 1 0 0 and minus 1 0 0 is 10 okay so assume that the point is x y z okay what are the distance of x y z from this point let's say I call this as p and I call this as a and I call this as b so as per the question p a plus p b is equal to 10 correct so what is p a p a is under root of x minus 1 the whole square y square z square p b is under root of x plus 1 the whole square y square per z square equal to 10 now should you leave the answer at this stage it would not be a good idea to leave it at this stage so what I'm going to do is I'm going to send this term to the other side and I'm going to square both the sides by the way I'm not writing everything I hope you'll be able to understand what do I mean to say so you all know that when I take it to the other side it will become something like a minus b so a minus b square is a square b square minus 2 a b okay lot of terms will get cancelled in fact this 2 will get cancelled this 2 will get cancelled x square and 1 will also get cancelled so you'll have a 4x minus 100 is equal to minus 20 this okay drop of factor of 4 I believe I can you can always drop a factor of 4 okay square both the sides so this will give you x square minus 50x plus 625 on this side you will have correct me if I'm wrong 25x square 25 y square 25 z square minus 50x and I think plus 25 okay again I think one cancellation is going to happen that is this guy will go off okay now collect every term to the right side so you will have 24x square you will have 25 y square you will have 25 z square and take this 25 to the other side equal to 600 okay so this structure is that of an ellipsoid it's a 3d structure which you learn in your first year of undergrad that is called an ellipsoid okay any questions here any questions in the process I think the process was quite familiar to you you have already done locus questions in 2d okay I'll ask you a very simple question find the distance of find the distance of 3 comma 4 comma 5 from y axis find the distance of 3 comma 4 comma 5 from the y axis done see again make a diagram diagram very very important see when you're trying to reach when you're trying to reach 3 4 5 what does it mean you go 3 units along x you go 4 units along y and you go 5 units along z isn't it that's how you reach 3 4 5 correct now what I'm trying to ask I'm trying to ask how far is this point from the y axis that means if I drop a perpendicular like this okay I hope my perpendicular is appearing to be perpendicular so let me make up perpendicular like this yeah if I drop a perpendicular like this what will be the length of this perpendicular now it's very obvious that you had moved three units in this way and you had moved five units in this way and this forms a right angle triangle so this distance P will be following a Pythagoras with this so your P will be under root of 25 plus 9 which is 34 units okay so please be careful such questions can also be asked in your school exam so with this I'm moving on to the section formula so section formula is basically a formula which will help you to find out in what ratio or what is the coordinate of a point which divides the join of two points let's say there's a point a x1 y1 z1 and there's a point B x2 y2 z2 so if point C divides it in the ratio of m is to n what is the coordinate of C okay how will I figure this out now all of you please pay attention there is a small construction involved over here so I'll I'll make my coordinate axes first I'm not going to draw a big one because it's going to cut the line actually so this is your z axis this is your x axis and this is your y axis now imagine that this line is in space I hope my oh my camera is off so yeah see it's in the space like this okay now take the mirror image of this line on the x y plane that means drop drop a perpendicular from both in fact all of these points that you have a C and B okay so what I'm drawing here in front of you is a mirror image mirror image is basically nothing but as if you are taking the shadow so let's say there's a light bulb okay so what is the shadow that this line will cast on the ground so basically let's say this was your line I hope you can see my my camera this was the line I have a light source on top it will cast some riff you know shadow on the ground right so this is that shadow I hope you can you know understand that this is the shadow so this is right angle okay let's call it as a dash c dash and b dash okay now assume that c had coordinates x y z correct now you tell me what would be the coordinates of a dash c dash and b dash you'll say so simple a dash will be x 1 y 1 0 because you're taking it on the x y plane this will be x y 0 and this will be x 2 y 2 0 correct yes I know understood so far so far everything is good okay now what I will do this reflection as it is I'm going to lift it up okay so this reflection as it is I'm going to lift it up and it will basically appear okay let me just use my gadgets over here so I'm raising this reflection like this so as it is this reflection line I raised it up like this so that this c dash again matches with the c so see let's say this was your line okay I hope you can see it on this camera correct okay on this particular hand of mine whatever was the shadow let's say this was the shadow I took the shadow up parallel to itself so this line gray line and this pink line are parallel to themselves so I lifted the shadow out till what height I lifted it till your c dash and c again combines now tell me how much is the height by which I have lifted it you'll say sir z isn't it so if I have written it by z can we write the coordinates of the pink line as this a double dash will be x 1 y 1 z and b double dash will be x 2 y 2 z do you all agree with me regarding the coordinates of a double dash and b double dash any questions okay now do you all agree that triangle a c a double dash will be similar to triangle b c b double dash that means these two triangles which I'm shading over here they are are similar to each other correct so they're similar to each other can I say a a double dash by b b double dash is same as ac is to cb and ac is to cb even though m has come up let me write it again ac is to cb is m is to n okay so this is m is to n correct now a double dash is nothing but this height correct and that height is nothing but z minus z 1 because they only differ in the z coordinates x 1 y 1 is the same for both of them similarly b b double dash is z 1 sorry z 2 minus z am I right so this height is z 2 minus z so this should be same as ac is to cb ac is to cb is m is to n now this is a very wonderful relation that I have to get the coordinates of z 1 sorry z in terms of z 1 z 2 m and n so on cross multiplication this is what I see let's club up all the z terms together and let's club up all the other terms together so z becomes m z 2 plus n z 1 by m plus n now this is just one third of your effort we can say similarly similarly had you taken the projection of this line on the y z plane you could have got x coordinate as m x 2 n x 1 by m plus n by the same logic and y coordinate if you had taken its projection on x z plane it would have been m y 2 and y 1 by m plus n so there you go you have the section formula for finding out the coordinates of the point which is dividing a line in the ratio of m is to n which is given by this now you'll say sir this is again similar to what we had learned in our 2d of course it is we are not learning something very you know out of the box it's just that a third dimension is added to the point and everything that you have learned for 2d will also be applicable like if you had a case of external division we will change this plus to minus okay we'll change these pluses to minus if there's a case of external division that is known to you one very important point I would like to highlight when c divides it's join off a and b whether internally or externally a very important phenomena that must occur here is a b c must be collinear there is no concept of there is no concept of internal or external division if the points themselves are not you know collinear very good question product there is nothing called slope in a 3d line there is nothing called slope in a 3d line so how do we know the direction of a 3d line that is obtained from a parameter which we called as direction cosines which is a subject matter of 12th so we'll not be talking about it in class 11 there are three terms which we call as lmn okay so unlike in 2d where only theta was sufficient enough because if theta is given to you the automatic angle was 90 minus theta but in 3d we need three dimensions or three you know parameters or three you can say informations to know its direction which are called direction cosines about that we'll discuss in class 12s don't worry about it okay so there's nothing called slope in a 3d line slope of a 3d line basically becomes direction cosines so that is the right word that we use meanwhile I would like to take a question on this very very small question this question may come in your school exams also the vertices of a b and c are as given to you the internal bisector of angle bac meets bc in d find ad so I hope the question is clear so let's say you have a b and c question is if you bisect the angle a internally okay it will meet at a point d and they're asking you the length of ad okay forget about length of ad find the coordinates of d only okay let's say I change my question I make it more simpler find coordinates of d okay because once you know d ad is a simple effort you don't have to you know break your head for finding ad now all of you here must be wondering what is the ratio in which d is dividing b and c I have repeated this in properties of triangles chapter when we were finding the in-center and I'm repeating it yet again angle bisector theorem bd is to dc will be same as ab is to ac I hope everybody still remembers this angle bisector theorem it was a class 10th topic correct so for that I need ab length okay let's figure out ab length ab length will be four square five square three square which is actually root 15 and ac length is a one square one square four square which is actually root 18 which is three root two oh by the way five root 50 is five root two okay so even though the figure is slightly skewed I hope the idea is clear that ab is to ac will be five is to three so this is five this is three okay so once you know the ratio you can easily find out the coordinates so the coordinates of d will be five into four three into one by five plus three comma five into three three into minus one by five plus three and five into two three into three by five plus three okay so this will give you 23 by eight this will give you 12 by eight 12 by it is by the way three by two and this is going going to give you 19 by eight so this will be your coordinates of d clear any questions one such question you may expect to come in your school exams also okay guys let me tell you from 3d geometry of 11th no j e questions will come why because this is just an introduction this is not even like you know five percent of the chapter five percent if you come it's like one percent of the chapter okay so j e may not expect questions to come from 11th grade 3d geometry right even in probability also don't expect lot of questions of 11th grade probability to come in j e may of course questions will come which will be you know taking your class 11th knowledge but per se don't expect that you know they will ask you questions on what are types of events and all in in j e may exam okay j e may has got a lot of serious affairs to manage they will not ask you those simple simple concepts the last topic that i would like to consider over here is the centroid of a triangle centroid of a triangle uh why i'm taking this is because many times in school they may ask you questions so what is the centroid centroid is basically the meeting point of the medians right which you all know know they are concurrent at a point which is called the centroid centroid divides the median in the ratio two is two one two is towards the vertex side two is towards the vertex side okay so what is the centroid of a triangle which is having the vertices as these points so again the section formula is going to help us uh let's say i call this point as m m will be the midpoint of b and c okay so g divides a m in the ratio of two is two one so g coordinate will be two into this which is x two plus x three one into x one so i will let it over here by two plus one two plus one is three similarly y coordinate also will be y one plus y two plus y three by three z one plus z two plus z three by three okay so this is the coordinates of the centroid again no difference from whatever we have learned in 2d is just that an extra dimension has come into picture now one thing that i would like everybody to know here i think you must be knowing also if you make a triangle by connecting the midpoints of the sides a b b c c a let's say m and p please note that g will also be the centroid of triangle m and p okay so not only a b c that will also be the centroid of the triangle m and p okay so this kind of concepts are normally asked so they will say okay uh the midpoints are known find out the centroid of the triangle so many people what they do they waste time finding a b c coordinates okay so they find a b c coordinates they spend around four to five minutes there and then they use this formula to get the centroid or you can directly get the centroid by finding the centroid of m and p also that will give you the very same result you don't have to find a b c for that okay so these type of questions are also expected to come in your school exam so this is all we have for introduction of 3d geometry there is nothing much in it okay i'll be sending you some worksheets some handouts on this also try to solve as many questions as possible okay so next class