Understanding Vector Spaces in 3D Geometry

hi uh it's time for another math easy solution and uh today we're gonna go over a new topic uh in my calculus book and that's gonna be a vector space in geometry and this in this first part we're gonna look at three dimensional or 3d coordinate systems so yes let's jump right ahead and uh note as always if you don't have time to watch this whole video uh you can play this video at a faster speed and here's a tutorial on that and also you could download and read the notes and you can also read notes on the hive blockchain and uh the yeah that link will be in the description below it's gonna be an article format on this hive blockchain and also you could watch a video in parts and the timestamps of all parts are in the description as well anyways let's jump right in so uh note the calculus book reference so note that i mainly follow along the following calculus book and that's calculus early transcendental seventh edition by james stewart and note in the earlier videos i used the sixth edition yeah so now i switched over to the seventh edition all right so here's the topics to cover so add these i'm going to add timestamps to these and you'll see that in the description below so uh part one here's vectors and the geometry of space uh three-dimensional coordinate systems and then we'll look at example one let's remove this period and then note on context example two example three and then distance formula in three dimensions then example four example five equation of a sphere and example six and an example seven. Alright, so let's take a look at this first part. So yes, the vectors and the geometry of space. So in this chapter, we introduce vectors and coordinate systems for three dimensional space. This will be the setting for our study of the calculus of functions of two variables in future videos. Yeah, we'll go over that in a later series, because the graph of such a function is a surface in space. so in this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space and examples of the surfaces and solids we study in this chapter are paraboloids used for satellite dishes so here's a parabola 3d parabola like this and so on so that's a 3d parabola there satellite dishes and hyperboloids are used for cooling towers of nuclear reactors notice this setup here is a hyperbola and then you have it in 3d and so on and here's that a link to different shapes here so here is a paraboloid here's a sphere and here's an ellipsoid so here's in 2d this parabola there's a circle there's an ellipse and there's a hyperbola then here's in 3d you can have this in separate parts like that yeah this one if you look at it this way if you look at it from that that way you'll have two parts like that and or you could have a hyper hyperboloid like this there's equations over them there's a sphere that's a problem so it was to continue further so uh three-dimensional coordinate systems so to locate a point in a plane or just a flat sheet so there's a plane like that to locate a point yeah like that uh what you'll need is well you two numbers are necessary we know that any point in the plane can be represented as an ordered pair so for example a b of real numbers where a is the x coordinate and b is the y coordinate. Now for this reason a plane is called two dimensional. To locate a point in space, so in three dimensional space, three numbers are required. And we represent any point in space by an ordered triplet or order triple a b and c of real numbers. In order to represent points in space we first choose a fixed point call this o the origin and three directed lines through o that are perpendicular to each other called the coordinate axis and labeled the x axis y axis and z axis but usually we think of the y the x and y axis as being horizontal and the zed axis as being vertical and and we draw the or the draw the orientation at origination So we draw the orientation of the axis as in the figure. So here's how it typically is shown. So there's the origin at the center there. And there's the y and x as the flat plane over there. And there's the z as the vertical over there. So these are the coordinate axes. And the direction of the z axis is determined by the right hand rule. And as illustrated below. So there's your right hand here. Notice the thumb sticks out. And typically you could also have the index finger stick out for the x axis. Notice it curls around here from the X to the Y and there's the vertical is your thumb. So if you curl the fingers of your right hand around the z-axis in the direction of a 90 degree counterclockwise rotation from the positive x-axis, from here to here is going to be 90 degrees like that. So this is a right angle and this is in other words 90 degrees like that to the positive y-axis. Then your thumb points in the positive y-axis. positive direction of the z axis so these are all positive axes like that uh the three uh three coordinate axis uh yeah the three coordinate axes or axis determine the three coordinate planes illustrated in the figure a below so here's the coordinate planes so this uh x y we'll call this the x y plane over here and there's the y z plane is vertical over there and on this side you'll have the x z plane on this side here so those are the planes there and uh here's added more text uh this is also for my calculus book i just forgot to add it but anyways here so the x y plane so this is the x y right here is the plane that contains the x and y axis so there's the x and y axes and the y z plane contains the y and z axis so this is z there's the y and the x y plane i mean you actually know not the x y the x z plane x z plane contains the x and z axes and these three coordinate planes divide space into eight parts so you have one here then you have another one in the back and here uh three four and then you also have uh on the negative sides so you're gonna have another five six uh seven and behind that's gonna be eight so you have eight parts total and those are called octants and the first octant in the foreground that's this right here this is determined from the positive axis those are positive ones over there and uh and note here so here we have a room there's a left wall right wall and there's a floor and that's because well many people have some difficulty visualizing the diagrams of three-dimensional figures so you may find it helpful to do the following and you can see this figure be above so look at the bottom corner of a room so let's say here call that and yeah and call the corner of the origin and then the wall on your left is in the x z plane so there's the wall there there's left wall and the wall on your right is the y z plane so there's the right wall and the floor floor is the x x y plane yeah and the x axis runs along the intersection of the floor and the left wall so there's the x axis and then the y is on the right wall there on the floor and the y-axis runs up from the floor toward the ceiling yeah yeah this is the oops actually no that's the z-axis it runs up from the floor toward the ceiling along the intersection of the two uh of the two walls the y-axis runs along the intersection of the floor and the right wall so there's the y-axis floor and right wall the z is in between these left and right wall goes up that like that yeah along the intersection of the two walls and you are situated in the first octant so you can you're inside this room right here the first one and you'll know that there's going to be well eight rooms total and you can now imagine seven other rooms situated in the other seven octants three on the same floor and four on the uh four on the floor below all connected by the common corner point here so if you're in this room there's gonna be another one here and then three and then over here and then at the bottom you'll have four more underneath you all right all right so now if p capital p is any point in space let a be the directed distance uh from the y z plane and yeah we have from the y is that plane to p and let b be the distance from the x z plane to p and uh this word directed i think the calculus just means uh perpendicular uh to it or the shortest distance from that plane and let's see be the distance from the x y plane to p you know and we'll illustrate this all soon and then yeah we present the point p by the order triplet a b c of real numbers and we call a b and c the coordinates of p and a is the x coordinate b is the y coordinate and c is the z coordinate thus to locate the point a b c we can start at the origin o and move a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the z-axis as in the figure. below so we'll put all these words into a drawing to make sense of it so you'll have this z axis like that the vertical one now you're going to have our y axis positive one here and this is the origin zero and there is our uh this is going to be perpendicular to it this is our x axis like that and let's say we have a point p in space here so it's in 3d we'll call this point p a b c all right so let's go make a sense of what this says so a is directed distance from the y z plane so if this is the y z plane is this set up across on this wall so it's a distance from that uh and that's again that's just our x coordinate in other words that distance is going to be up to here somewhere or uh yeah so it's going to be up to here but before that so let's say we have this i'm going to draw this all the way down here all the way to the bottom and then um yes let's say it hits the bottom here and then draw this perpendicular it's going to go all the way across here like that oops and this is parallel to this one right here so this is parallel to this wall and this one here is parallel to this like that yeah so then this distance from the z y plane is the x coordinate and in other words we can start from the origin zero and move a units along the x axis so there's the a units and then b units parallel to the y axis that's b like that and then c units parallel to the z axis like this so b is going to be this distance b is it is the distance from this z x plane or the x z plane x z plane is b and then c is the distance from the x y plane to p so is the x y plane and there's a z and this is c not z like that and see this is all the way up to there all right so going further so the point a b c determines a rectangular box as in the figure below so so we have this right here let's draw this again but now you can see that that this can form a rectangular i mean a rectangular box so we'll do that again so we have the zed like that and then we're going to have our y i'm just going to make this yeah it's a bit shorter so this is our y and then this is going to be our x like this oops this is going to be our x like that and let's say we have our point p across here so this is our point p and we're going to have a box so let's draw this all in 3d this goes down here this goes draw this a bit neater so this is let's say it goes here and there is the line across there's the line across and then draw this all in 3d like that. And then this goes across like that. This goes like that, there is our 3d box. Alright, so there's our box. And this one, we're going to call this P, A, B, C. Alright, continuing further. So if we drop the vertical perpendicular line from P to the x y plants, if we drop this line, just remove that in other words, our vertical is in C becomes zero, we get a point q with coordinates a, b and zero called the projection of P onto the x y plane. So that's this point right here, this is our point q like that. And we'll call this a, b and then the vertical distance is zero like that. And similarly, r zero b, c and r the points are a zero a a 0 c are the projections of p onto the y z plane and x z plane respectively so in other words this one we're going to remove the uh this a value that's this uh horizontal x coordinate from here to here so in other words we're going to get this point here so actually no this one here we're removing it so we get this point here this one's going to be our r and then the x coordinate to zero everything else is the same b and c like that and that's projection onto the z uh onto the y z plane and that's right here y z plane and then the next one is our s we're going to drop this down yeah we're going to drop the y coordinate so we're going to project it all the way onto yeah onto this uh x z plane so the x z plane so we're going to drop this line and we're going to go all the way up to here so this is p q r and this one is Now let's put this like this. This is going to be our s. This is going to be our a. 0 is the y-coordinate, and then this is going to be c. Like that. all right continuing further so as numerical illustrations uh the points right here negative four three negative five and three negative two negative six are plotted in the figures below so let's graph this one or let's plot that point out first so this one is notice it's going to go uh negative four so we're going to go the negative x axis then it's going to be positive three and then this is negative so in other words it's gonna be on it's going to be on the below these uh axis this is the z it's going to be a negative one so let's say we have our y like this and we're going to have to extend this all the way down because it's going to be negative z just extend this down further and we're also going negative uh negative x so that's going to go backwards so we're going to have to go from here this is our x but then we're gonna have to go to the backwards side of it so we're going to go negative four so let's uh put it all the way here so we're going to go negative four and let's put that line here and then we're going to go uh it's supposed to be all the way up to here and then we're going to go uh plus three to the y make it a bit shorter than that so let's make it um here let's go uh okay this should be good all right so this is good enough so there's three right there and then we're going to go all the way down uh we're gonna go all the way down to negative five so let's say the point is all the way across here we're gonna be drawing in 3d this goes it's going to go behind it so i'm going to break that apart and yeah here so this is going to be negative 5 all the way down and there's our point negative 4 3 and negative 5. and now let's draw this in 3d yeah so to do that let's draw our top part here and draw a parallel line like that and this is going to go all the way down and this more down like that until this is parallel like that and then we're going to have this line across here parallel to that a y-axis and then behind it we're looking to have dashed lines dash dash dash dash dash dash and then this is going to be another dash line like that and then dash line like that all right here so i fixed that up and made it smaller just so we can uh fit this inside it will fit the other one inside so this one is going to be 3 negative 2 and negative 6. so it's also going to be below this uh it's going to be below this floor because this negative vertical it's going to be yeah it's going to be negative six there but now it's gonna be in the front and then we're gonna go backwards there so let's draw this out so we're gonna do again the zed axis but it's going to be on the negative y so we're going to have a short z we're going to have a short y like that and then it's going to be on the positive x so we can make this super long there's the x like that and but it's going to be on the negative y so make it like that so negative y so we're going to go to 3 here let's say all the way to 3 here and then we're going to go negative 2 we're going to go all the way across here negative two and then we're gonna go all the way back down all the way until negative six so we're gonna go all the way down and we're gonna go like this negative six like that and then draw this in 3d and well this is the point right there and this is going to be our uh three negative two negative six like that and then we're gonna draw this all in let's make this bigger and now we're gonna draw in 3d so it's gonna go like that So it's going to go all the way down until it's parallel. Well, this one here as well until it's parallel. So all the way down like that. And yeah, this one across like that. So make this all the way down like that. Yes, it looks something like that. Now we draw this in 3D behind it. So this is going to be all the way down. Dot, dot, dot, dot, dot. And we'll be down like that somewhere. And this is going to be dot, dot, dot, dot, and then and yeah here i just fixed it up made a bit smaller to match this and yes it looks like that in our rectangular box there oh anyways let's continue for this so the following cartesian uh product is the set of all order triples of real numbers and is denoted by this r to power of three and this is the latin capital r usually written like that which i sometimes just use r cubed there and this cartesian product this is just written as R, Osbertus in the Latin form. R X R or times R times R and this equals to the set related like this of all these points x y z where x y and z are elements of real numbers in other words these are all just real numbers so this is written like that so this is the set where x y z these are all just real numbers or it's a collection of all the points and it's for completeness so this r times r times r this is just indicating this is our r cubed equals all this all right so now that we have this this is just a set of all of these points x y z are of all these ordered triples and we have given a one to one correspondent correspondence between points p in space and ordered triples a b c in r3 where these are just all real numbers inside here so each point has a one-to-one correspondence with uh the uh yeah with with these collection of triples right here this x y z and so on for this a b c a b c and r cube so this is called a dimensional rectangular coordinate system and notice that in terms of coordinates the first octant can be described as the set of points whose coordinates are all positive remember yeah that's where all the positive axes are uh in two-dimensional and analytic geometry the graph of an equation involving x and y is a curve in r squared yeah curve or just a line like that's just a curve but in three-dimensional analytic geometry an equation is x an equation in x y and z represents a surface so in 2d it looks like that in 3d it's going to look like yeah just an actual surface imagine this is in 3d so a full surface like that and we can illustrate this in example one so example one what surfaces in r3 are in this set of all real triples right here are represented by the following equation a which is z equals to three and b which is y equals to five all right so let's take a look at the solution to part a this is z equals to three so the equation z equals to three represents the set it's all the points x y z where z is equal to three and which is a set of all points in r3 whose uh z coordinate is three and this is just a horizontal plane that is parallel to the xy plane and three units above it as in the figure below so the equation z equals three in three dimensions that's where all the coordinates are have a z equals to three variable or a value so if we have this let's just draw our uh x our z axis like that this is going to be our y axis or let's make it a bit more 3d so this is our y and then this is going to be our z i mean our x like that and this is our origin at the zero here this is our origin and now we're going to go all the points that are z is 3. so in other words we have to go all the way up to here three so here's a three point but it's a plane number so it's going to go to infinity everywhere so it's gonna let's just draw this in 3d so it's going to look like this and all the way across like that and uh since this is in front of it i'm just going to remove this remove this vertical i'm going to put this like this so this is above and now it's going to go below dot dot dot dot dot dot better so that's at the three this is going to be dot dot dot so that's the axis is behind it so we're drawing drawing this in 3d so it looks like that there and we could write here as this is z equals to three and this is a plane in three dimensions or r3 like that so in r3 So yeah, typically when I'm writing this by pen, I'll write it like this in Latin form, but when I type it out, I typically write it like this because it's easier, because there's no button on my keyboard for the Latin version. Alright, so now let's take a look at the solution to B. So B was this equation, y equals to 5. So if z is equal to 3 as a horizontal plane, then y equals to 5 as well, it's going to be a vertical plane. So the equation y equals 5 represents the set of all points in r cubed whose y-coordinate is 5. This is the vertical plane that is parallel to the xz plane and 5 units to the right of it as in the figure below. So if we draw our... this is our Z axis and we're gonna go across here, so that's our Z, I'm gonna put it something like this, there's our Y, alright let's make it even further, this is our Y and this is gonna be our X like that and this is where Y equals to 5, so let's go all the way up to here 5 and this is going to be a plane, so we're gonna draw a line like this line all the way down it's going to be behind it draw this in 3d just to make a just a giant plane like this like that and yeah as before here just fix it oh well made it like this and as before we got to draw the dash lines for the ones behind it i'm going to erase this right here draw this across there and then it's going to be dash line so this one's going to be straight across and it's going to be dash dash dash like that yeah so we have that line like there and also we could shade this in like that shade it in so it's a plane and shade this in like that and yeah so we have this equation is y equals to five and this is a plane in so a plane in r3 let's try this better All right, so now that we have that, that's a vertical plane right there. But now let's take a look at just a quick note on context. Because this one here, we're asked to solve equation y equals 5 in R3. But if we're looking at two dimensions, so when an equation is given, we must understand from the context where it represents a curve in R2 or a surface in R3. So in example one, y equals 5 represents a plane in R3. But of course, course y equals 5 also represents a line in r2 if we are dealing with two-dimensional analytic geometry as in the figure below so if we have uh there yeah this is just our x y axis this is our x this is our y then the line if this is our origin zero then the line is just like this this is just y equals uh this is the 5 in other words this is y equals 5 and this is a line you and r2. Alright, so now let's continue further. Now this section is out of the title here. So equations of vertical or horizontal planes. So in general, if k is a constant, then x equals k represents a plane parallel to the y z plane. Yes, in other words, where x is fixed, so that's going to be a vertical Yeah, this one's gonna be a vertical uh plane and then y equals k is a plane parallel to the x z plane that's also vertical exist as a z then z equals to k is a plane parallel to the x y plane and that's gonna be horizontal and so on so in the earlier uh rectangular box figure above actually before we get that i'm just gonna write over here represents a vertical plane a vertical plane and this one's uh whatever z is constant horizontal plane parallel to the xy so in the earlier rectangular box figure the faces of the rectangular box are formed by the three coordinate planes x equals zero this is the y z plane y equals to zero this is the x z plane and z equals zero is the x y plane and also these right here x equals a y equals b and c z equals c and we can see this if we just scroll up to that earlier a rectangular box we had So that's this part right here. And what I'm going to do is I'm going to add all the points here. So this is B, if we remove this A, it gets to this point, we'll just call the zero, B zero. And then this point across here, this is our zero, I know that's going to be actually this is going to be a zero. So we're going to remove this B there. So it's gonna be zero, zero, C, and also put this into the origin. and then we have this part right here yeah this part here is going to be our this is when we take this a right here and then but then we just remove the b is zero so we're going to get over here it's going to be a zero zero like that all right so here i just teleported above back down here and i just just added here so that same figure went over copied below so we copied here made it a bit smaller and now we could look at these six planes there. And this top part is where the Z is constant. actually before we get to those let's go these this X equals 0 Y equals 0 and Z equals 0 these are the yeah those are the three coordinate planes here let's move it over here so we have the three coordinate planes and then we have also these extra planes so if we look at the coordinate plane I'm going to draw this in red where we have X equals to 0 let's see why is that plane Y Z that's in the background here let's make this a out in front of it so like this like that yeah and then etc so that is uh this part right here this is our x equals to zero like that and then the next one is let's make a different line here so this is this where y equals zero that's this can be x z plane this is y equals zero like that that's this or y equals zero like that and then this is a z equals zero is at the bottom where yet z is completely flat there the bottom like that and so on that's our z equals zero and then let's draw a different color it's a bit hard to see but just to illustrate and now this is the x equals a is when you have it all the way across this this face this is our x equals to a and then uh y equals to b is all the way across this face here and uh yeah x equals to b like that yeah so it's a bit uh messy there then we have z equals to c that's a that's a the top of this box but again z equals to c like that All right, yeah, so it's a bit messy there, but I think it's, if you were watching the video, you could follow along, and yeah, just follow along how I did, but if you're just looking at the notes, it's going to be a bit more complicated, but anyways, it just looks a bit messy, but I think it's good enough. All right, so now let's continue further, and I'll look at example two, and this one has two parts, part A and B, and part A states which points X, Y, and Z satisfies the equations. x squared plus y squared equals 1 and z equals to 3 and then part b says what does the equation x squared plus y squared equal represent as a surface in r3 all right so let's take a look at the solution to part a first so because z equals to 3 the points lie in the horizontal plane z equals to 3 from example 1a so if we recall example 1a that this is our z equals 3. so it's going to be somewhere along this there but then we're also given an equation of a circle this is just equation of a circle so because x squared plus y squared equals one the points lie in the circle with radius one and the center of uh yeah and the center on the z axis as shown in the figure below so we draw this out we're going to have something that looks like this this is our z and there's our y axis our x axis there's the origin and this is our point three and and now we have it like that i'll make it a bit shorter then we have a point three here and now we're now so that's a three on the z but now we have x squared plus y squared that's the circle on this uh z equals three plane so we can write it like this right like this but there's a circle and uh yeah yeah that's just a circle there and well it's going to be behind this so let's make this like this dash let's try this one more time all right yeah so just have it like that so this is going to be our circle yeah so it's going to be circle and the equations are circle x squared plus y squared equals to 1 and z is equal to 3. all right so now let's take a look at the solution to b so b is here so we're given what does the equation x squared plus y squared equals 1 represent as a surface in r3 and in other words in three dimensions and this is the same thing as this but there's no z equals three so let's take a look at the solution to b so given that x squared plus y squared equals one with no restrictions on z we see that the point x y z could lie on a on a circle in any horizontal plane z equals to k so the surface x squared plus y squared equals one and r3 consists of all possible horizontal circles x squared plus y squared equals one and z equals k and is therefore is therefore the circular cylinder with radius one whose whose axis is the z axis so basically we're not limited to the z equals three it could be on anywhere across i'm just going to copy and paste that same graph here yeah so i'm going to copy and paste that and now we can remove this three so it does not have to be the three and it could be anywhere and this is going to go all the way down yeah we'll just go down here it's down here i'm just gonna i'm just gonna make it on the uh the positive axis but it can go all the way down and this is gonna be a cylinder like that a bit better like this and and then this is gonna be in 3d so i'm gonna erase this put this down here and then dot dot dot all the way down i'll actually keep going because it's a it's a surface not a 3d shape so i mean not a solid so it's not a solid so then put all the line all the way down there and go dash dash dash same thing with this dash dash dash dash dash extend it out just to make it 3d dash dash dash dash dash dash and then like that and i'm just going to shade this in blue just make it like this or actually in red shaded shaded cross or it's a bit hard to shade shade shade and then the inside shaded as well like that yeah just do that just to show it's a surface and then basically we have surface uh this is the cylinder or this is a cylinder right here all right so those are cylinders just typed it a bit better so x squared plus y squared equals to one and uh yeah so the radius here yes to the radius across here is one, the articles one like that, or instead of making it this way, this right articles to one here, articles one, yeah, from there to here. So these are going to be one, one, so. All right, so now let's continue further. And now let's look at example three. And this one states, describe and sketch the surface in R3 represented by the equation y equals x. So let's take a look at the solution. So the equation represents the set of all points in R3 whose x and y coordinates are equal. That is, if we write this set out, we'll write it like this. So this bracket x, and then x. y is x, and then z like that. And again, this is because y is equal to x. So we'll write them both as x. And this is where x and z. you could write it as x z are all real numbers or you could even write it as x is an element of all real numbers like this and z is an element of all real numbers like that just calculus book wrote this method or in this form but you could write it as x and z as all real numbers or x uh all real numbers and z all real numbers like this and then just close bracket like that Now, this is just a vertical plane that intersects the x, y plane in the line y equals x and z equals to zero. And the portion of this plane that lies in the first octant is sketched in the figure below. And because it's a diagonal line, I'm just going to draw this out like this. The calculus book draws it. So it has the z-axis like this. Here's the z-axis. So that's the z-axis there. And then it has the... x-axis over at this side let's fix it up so x-axis over there and then it has the y-axis over like this y and then I'll put the origin here and that's just so that we could have this line better presented so I'm going to cross this line here so this is at the y equals x line and that z equals zero and then now we could draw a vertical line across I'm just going to make it shorter and then it's going to be like this it's shorter like that it's going to look something like this make it vertical and parallel and this is going to be like that and again because this is in 3d let's just erase this or we could do just move it up like this actually it's better just delete it and then start fresh so it's going to be like this this is going to go across like that this will be dash dash dash dash dash dash dash like that and then again if we just shade this out in red like that so that's the plane like that and this is at the y equals x in other words at the perfect center there so the same distance across and we're going to write this as a plane uh the yeah the plane y equals to x like that all right so now let's continue further now let's look at uh distance uh formulas well first let's look at the distance formula in 2d and then we're going to extend it to 3d so recall the 2d distance formula between two points and basically what we have is like this so let's say we have y x y axis just to derive it here and then we have a point here this is point one p one and this is our x one yeah call this i'll call this p one x one and y one then let's say we have a point here p two this is our x two y two like that So the distance between this and this, call this absolute value p1, p2, like that. So that's a distance. And then we could use Pythagoras, draw a right angle triangle. And then this distance here is going to be absolute value of y2 minus y1. This is absolute value of x2 minus x1. And then Pythagoras. Yeah, and here I just typed it out here or just wrote it out. So then we just apply Pythagoras or Pythagorean theorem. And then square root. we're going to get p1 p2 is equal to the square root and we'll only look at the positive square root is going to be square root over this one here let's put that for completeness well yeah if this is absolute value squared it's the same thing as writing x squared minus x1 squared plus y squared minus minus y1 squared like that whether it's absolute value or not when you square it's going to be positive it's going to always be positive so let's just uh just recall this formula right there all right so now that we have this and now the familiar formula for the distance between two points in a plane this one above here is easily extended to the following three-dimensional formula and here is the distance formula in three dimensions yeah so we have the distance p1 p2 between the points p1 x1 y1 z1 and p2 you x2, y2, z2, this can be extended over and it's going to be just p1, the absolute value of p1, p2 is the distance, this equals to square root and this just extends it over x squared minus x1 squared plus x, I mean plus y now, y2 minus y1 squared plus z2 minus z1 squared. Alright, so now that we have this, and then we're going to go over a proof of it. So proof to see why this formula is true. So here, we just extended it and added the Z component here. See why it's true, we construct a rectangular box as in the figure below, where p one and p two are opposite of vertices, and the faces of the box are parallel to the coordinate planes. So let's draw this all out. So if we have a z axis like this, and there's a y axis like that and this is our z like this and then this is our x like this like that and let's draw this uh just in the top here just to get it high up so it doesn't interfere with the axes so let's say this is our point um this is our point p2 and let's say we have a point across uh here well i'm just going to draw a box first so let's say we have a box like this and just move it even here so the box like this and then this goes down like this this goes like that so we have a box and this is going to be parallel parallel until we have it vertical like this and then we have this across like that and then behind it, all the way to the bottom, this is going to be all the way behind it. All right, so we have this, make this a bit neater, or downwards more. All right, and at this point here, we'll call this P1. This is going to be P1, P1, and this is going to be our x1 actually before we get that i just move this over here so we have some more space we're going to have p1 this is going to be our x1 y1 z z1 like that and now this point right here we'll call this p2 like that and erase this and draw a bit bigger okay so this is p2 p2 is our x2, y2, z2, like that. Alright so now we have this and what I'm going to do is I'm going to consider this point here as B. This is our B point and this is going to be as x2, so all we're going to do is drop this z all the way down until it matches the z1 there. y2 and then z1 like that and this is I'm going to draw a line across here, dash line across and this is a perpendicular line like that. I'll just fix that up it's going to be like this perpendicular line and and now we're going to need this point across here from here to here this is the point that we want so this is um i'm going to write this as i'm going to write this as a distance this is going to be p i'm going to write this in red so all the distances i'm going to write in red p1 p2 absolute value all right so this point is b and now this point here is our a and i'm going to write this a as a is x two so basically it's going to be the same thing at this point here but we're going to remove this yeah it's going to be same as this one but we're going to remove this x1 and replace it with x2 so it goes over to here this is x2 this is going to be x2 and then everything else the same this is going to be y1 z1 like that all right so now we have this a b like that I'm going to write this as absolute value of a b like that and again this is uh i'm going to write this here this is another right angle so we have two right angles across here and there all right so we have a b there now we have this part here from the top to bottom this is uh we'll call this as p2 b like that and the next one here is from here this a to p1 at the bottom there that is going to be our absolute value of p1 and then a and these distances this is just well this is a horizontal distance remember it's parallel to this and and that just means well it's going to be x2 minus x1 the absolute value of it so we could write this all down so you could write absolute value of p1 a so this distance is going to equal to absolute value of x2 minus x1 and then this one right here ab this is parallel to this y axis so this distance is going to be the y's so then we have a absolute value or the distance a b is equal to absolute value of y y1 y2 minus y1 and then the last one is this vertical component it's just vertical parallel with this and so that's just going to be z2 minus z1 and that's our p2b like that so we'll write this as p2b is equal to z squared or I'm z2 minus z1 absolute value all right so now that we have this set up well we notice what we have is well we have two right angle triangles so we can basically find this distance which is our p1b and then we can solve this distance up top which is p1 p2 so we have two triangles there and I'll just change this back to red so we have this distance right here p1 and then b like that so p1b is down here so that's the hypotenuse of this triangle and then it will become the base of this other triangle that's the hypotenuse so we can apply pythagorean theorem twice so because triangles p1 b p2 so p1 b p2 and p1 a b so p uh yeah p1 a b right here our right angle two applications of the pythagorean theorem give and yeah let's just write it out so let's write our distance p1 uh b2 oops this raises so p1 b2 there so we're going to have uh p so absolute value p1 uh b2 and i'm going to square it just because pythagorean theorem is squared is equal to p1 b plus p2 b squared there so this equals two uh p1 b squared plus p um this is going to be uh p2 b squared like that so that's this part right here this and this is this over here that's what we have there and then then then the second triangle is this is the hypotenuse and this hypotenuse square is equal to this squared and this squared so in other words this part right here is going to be you p1 b squared is equal to p1 a squared plus a b squared and again this is just this part so in other words we can combine we could throw this inside so throw this inside here so that's what we can put inside and when we do that we get here so this becomes p1 p1 p2 squared is equal to and then we just replace that replace this with this p1 a squared plus absolute value a b squared plus the distance p2 let's play this better p2 b squared and we know what these all equal those are just these p1 a1 is x squared minus y minus x1 i'm an x1 i'm an x2 minus x1 not x squared and then a b is here y2 minus my one and then a p2b is this one here z 2 minus that one and that's all squared so we could write this as equaling to absolute value x squared minus x1 squared plus y2 minus my one absolute value plus z2 minus z1 squared and since we're squaring it's going to be positive so we could put this all together so thus and then and also square square rooted just to get it all out of the way and look at only the positive of the square root of p1 p2 is equal to square root and write this as x and then just write it in normal bracket x2 x1 squared plus y2 minus y1 squared plus z2 minus z1 squared and we can just box this all out like that so there is our distance formula in 3d All right, so let's just quickly double check. And if we scroll up here, yeah, so this is the one we were given. This is, yeah, the distance is just x squared minus x1 squared, and then plus the y's and z's, and then all squared. So exactly the same thing as this right here. Now let's take a look at example four. And this one doesn't really give a question, just because it's pretty straightforward. So the distance from the point P to, yeah, from the point P, which is two, the coordinates are 2, negative 1, 7. to the point q which is 1 negative 3 and 5 is so we just apply uh this formula there so pq is equal to square root and now the x x 2 minus x 1 so that's going to be 2 minus 1 2 minus 1 squared then plus y the 2 minus 1 so we'll just have this negative 1 minus 3 We're actually just for completeness because this is x2 minus x1 just to make it the same form. It doesn't matter because you're squaring it and it's always going to be the same. It's always going to be positive. So let's just go 1 minus 2. Let's go from this as our x2 or our second point p2. So then it's going to be negative 3 minus 1. Negative 3 minus 1, that's going to be, I mean minus negative 1 is going to be plus 1. And then plus the next one is 5 minus 7. 5 minus 7 squared and now this equals 2 and well square root 1 minus 2 is negative 1 but you're squaring it and it's always gonna be positive just go 1 squared plus this one here negative 3 plus 2 or I'll just for completeness I'm just going to keep this in negative 1 squared plus and the negative 3 plus 1 is negative 2 squared and then plus 5 minus 7 negative 2 squared this equals 2 square root and then this is going to be 1 negative 1 squared is 1 negative 2 squared is plus 4 and then the next one negative 2 squared is plus 4 again that's going to be 8 plus 1 is 9 equals to square root 9 equals to 3 yeah 3 or or you could even have a negative 3 so negative 3 times negative 3 is also put that for completeness plus minus but it's just going to be that we just look at the plus side of it yeah so just uh just erase this so just do a complete this because when you're square rooting it's uh you can have still negative three times negative three but you can't have a negative distance in this case because we're defining it as absolute value all right so now let's take a look at example five and this one states uh find an equation of a sphere with radius r and center h k a 1. actually not one it's actually l so kaku's book writes l lowercase but it's it's just uh for me i like using capital case i guess because it's easier it doesn't look like a one because this one here i write one like that sometimes so this is how it writes l lowercase and i'll just write it as capital l uh so find an equation of a sphere with radius r and center uh c which is coordinates h k and l so solution by definition a sphere is a set of all points p uh x y z whose distance from c is r and you can see the figure below so let's just draw this in 3d so sphere is here's the z axis like this and there is the y axis uh this is our y and there is our x and there's z like that and this is we have the center here you the center inside is let's say c c and this is h k l and then the distance across all the way to because we're inside a sphere to p of x y z like that it'd be like this and then it's going to be always the same it's going to be r like that and just draw this like that and I'm just going to draw this, shade this in red. So it's a 3D sphere, shaded in like that, something like that. So it's a sphere. All right, so we have this sphere like that, and the distance from C is going to be always, yeah, it's always going to be equals to r. And here I just move this over to make it more in the center there like that. All right, so that's actually in the center now. of the sphere so thus p is on the sphere if and only if the distance pc equals to r and then if we square both sides we have we're going to have pc squared equals to well r squared and this equals 2 and the number of distance formula is just uh we subtract the x's and the y's and the z's and so on and squared and because we already uh we're squaring that so we get rid of the square root this becomes x minus, so this is the x, this is h, this k is the y and so on, x minus h squared and then plus y minus k squared plus z minus l squared. Yeah so thus our equation of a sphere with radius r and center c hkl is just this, so we can box this r squared over like that, so there's the equation of a sphere. all right so now that we have this and uh and basically this result of example five is worth remembering so we'll call this equation of a sphere so an equation of a sphere with center c hkl and radius r is and we'll write r squared on the other side so we're just going to write this as x minus h like this squared plus y minus k squared plus and then z minus l squared equals to r squared like that all right so now uh in particular if the center uh is the origin uh o then an equation of the sphere is so this is uh the c is all going to be zero zero zero that just means that the equation is just going to be x squared plus y squared plus z squared equals two r squared again uh compare this with the equation of a circle which is just x squared plus y squared equals to r squared all right so now this brings us to example six and this states show that the following is the equation of a sphere and find its center and radius so we have this equation x squared plus y squared plus z squared plus 4x minus 6y plus 2z plus 6 equals to 0. so solution uh we could rewrite the given equation in in the form of an equation of a sphere if we complete squares the first recall completing squares recall yes if we recall right here x plus a squared yeah this equals two because that's just a square so this equals two well x plus a times x plus a so we're going to multiply x by x and then x plus a and then x by a then a times x then a times a in other words we have twice we have an a times x and then x times a so this equals to an x squared plus 2a and then the last one is a squared plus a squared so x times x x squared then uh x times a plus a times x is 2a yeah 2a x like that let's erase this and move it uh 2a x and then the last one a times a is a squared like that yeah so the the key thing is we have this 2a and then we have this a squared there and that's what this x plus a squared is becomes this yeah so thus we just need to write these x and these y's and the z's all in this form so we can complete the square and what we'll do is now so thus so we have this i'm just going to blank this out thus and let's just move over this x over here just for completeness let's write it all out uh this is x squared and then plus at 4x so we're going to write this as uh let's write it uh this is right here x squared plus 4x put a bracket just to separate it and then the next one is is this uh this y and there's a negative 6y so we have a plus y squared minus 6y and then the next one is uh this is going to be plus z squared and then plus 2z like that plus 2z and then this 6 we're going to move it over to the other side it's going to be negative 6 like that and now if we look at this completing the square so this part right here there's a 2a there so all these have a 2ax to 2ax but we need to add a squared so in this case here we have 2a equals to 4 so that means a is equal to 2 and a squared is equal to 4 this means yes this means a squared is equal to 4 i'll also remove that arrow so we have a squared is equal to four this way it's small and now this case here we have so this right here this is put an arrow here all right so go back to here so this negative six that means we have our two a is equal to negative six that means our a is equal to negative three and then our a squared is going to be well positive nine like that and the next one is over here our 2a is equal to 2 this means our a is equal to 1 and a square is also equal to 1 like that yeah so what this means is well we need to add an a square to each one and then also subtract one from each equation so that we don't change it and then move that that additional a squared over to the other side just so yeah let's put it all over there so what we'll get is so we get x squared plus 4x minus i mean yeah this is going to be plus a squared now a squared is 4. so that's our a squared forward like that next one is and we're going to subtract by 4 squared as well this next one's going to be y squared minus 6y then plus a squared which is 9 plus 9 we're going to subtract that as well and then plus the next one is z squared plus 2 z i'm just going to put this here a squared for completeness plus 2z and this is going to be plus a squared it's going to be 1. i got and then at the right side is negative 6 so we want all these extra stuff that we added to cancel so we just add these on this side as well it's going to be plus 4 and then minus uh yeah i mean yeah this is gonna be plus four plus nine because they cancel we make them the same sign because it's on the other side and they would cancel yes we're not changing anything and then the next one is plus one like that and let's just add these up so negative 6 plus 4 is negative 2 like that and then these two is 10 so then negative 2 plus 10 this is just 8. yeah all right so this becomes and now this is completing the square so x squared plus 4x plus 4 that just becomes x plus a squared where a is 2. so this becomes x plus 2 squared like that next one's going to be uh this next one's going to be y this one right here a is negative 3. so we're going to have a y oops and go back here so y minus 3 squared and then plus the next one is so we hear this square completed uh this a is just one so we get x plus z plus one like that z plus one squared and this all equals to 8 like that so this equals to 8 and remember this equals to R squared and this whole thing is just an equation of a sphere, equation of a sphere, all right, so thus, and then we're asked here, so we're asked to find its center and its radius, so this is r squared, so thus our center, I'll write thus, the center is equal to, and remember it's going to be the negative of these, so this is x plus 2, uh it's going to be the negative because if we scroll back up to our equation here negative h the center is h negative k center is k and so on so we do the negatives so center is going to be negative 2 plus 3 so it's right at 3 and then negative 1 like that and the radius is i'll write radius radius so make it a capital actually the center and then radius ray d is is going to be uh r squared i mean r is going to be well is equal to square root uh yes square root 8. so here we square root 8 and this equals 2 which simplified calculus books usually does that 4 times 2 so the 2 becomes and the 4 becomes 2. so 2 times square root 2 and we're looking only the positive of the square root so that is the radius and we could just box out the center and the radius all right so now let's take a look at the last part of this video and that's example seven and this one states what region in r3 is represented by the following inequalities and we have this one is less than or equal to x squared yeah x squared plus y squared plus z squared and is less than four And we're also given that z is less than or equal to 0. So let's take a look at the solution. So the solution, the inequalities can be written as, and basically we're going to take these two inequalities, these two, this is greater than 1 and less than 4, less than or equal to 4 and greater than or equal to 1, and write this, and then basically square root both sides. So what we'll do is 1 square root less than or equal to square root x squared, plus y squared plus z squared and this is less than or equal to four i'm squared four this equals to two so what we'll do is uh this basically equals to just one less than or equal to two yeah less than equal to uh squared x squared plus y squared plus z squared in other words that's a distance less than or equal to two and we'll only look at the positive because we're looking at distance right here And yeah, these inequalities right here represent the points x, y, z, whose distance from the origin is at least one, that's greater than one, and it's going to be at most two. But we are also given that z is less than or equal to zero. So the points lie on or below the x, y plane. Thus, the given inequalities represent the solid region that lies beneath or on the xy plane and between or on the uh the following spheres pause right over the spheres and we're going to write this as x squared plus y squared plus z squared equals to one and in between these two right here and x squared plus y squared and it's just equation of a sphere plus z squared and then the r squared uh this is going to be r squared is just well two squared plus four and uh yeah so now if we were to draw this out uh this is going to be let's draw it like this so this is our um let's say i'll draw our z axis like that and then our yeah our y axis so that's a z and we're going to draw our y axis going across negative and positive there's the y like that and now we're going to do the x axis across here and there's our x axis and now what we'll do is well it's going to be two spheres distance one this this is the origin and we're just going to have a one here i'm going to draw a sphere here with distance one or circle across this well actually because uh it's going to be behind that so i'm just going to write it like this behind it and be on these be like that. Alright, here, I just fixed it up made it like that. And the next one is going to be a distance to be like this. and again it's behind that z-axis all right here i just fixed that up so it looks like that and then it's again it's a solid region so let's write this let's draw this out like that it's going to be underneath this and then it's going to be behind this like that and now let's shade it in shade it up top here so make it look like it's 3d and shade it in etc whoops or actually i'll just shade it in like that so it's shading it in and now here fix up the shading like that so so yeah that's uh what it has just made it look like so it's an indented inside and there is our uh yeah half of a uh half of a sphere with a hole inside but then it's cut off so it's half of that yeah essentially like a thick coconut here that's cut in half oh yeah coconut with a thick skin there and uh yeah pretty interesting stuff and that is all for today and yeah so hopefully you enjoyed I'm just going to scroll back up and yeah I'll be continuing on this series in future videos on this vectors and geometry and 3d geometry and so on anyways that is all for today thanks for watching and stay tuned for another math easy solution

So in this chapter, we introduce vectors and coordinate systems for three dimensional space. This will be the setting for our study of the calculus of functions of two variables in future videos. Yeah, we'll go over that in a later series, because the graph of such a function is a surface in space. so in this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space and examples of the surfaces and solids we study in this chapter are paraboloids used for satellite dishes so here's a parabola 3d parabola like this and so on so that's a 3d parabola there satellite dishes and hyperboloids are used for cooling towers of nuclear reactors notice this setup here is a hyperbola and then you have it in 3d and so on and here's that a link to different shapes here so here is a paraboloid here's a sphere and here's an ellipsoid so here's in 2d this parabola there's a circle there's an ellipse and there's a hyperbola then here's in 3d you can have this in separate parts like that yeah this one if you look at it this way if you look at it from that that way you'll have two parts like that and or you could have a hyper hyperboloid like this there's equations over them there's a sphere that's a problem so it was to continue further so uh three-dimensional coordinate systems so to locate a point in a plane or just a flat sheet so there's a plane like that to locate a point yeah like that uh what you'll need is well you two numbers are necessary we know that any point in the plane can be represented as an ordered pair so for example a b of real numbers where a is the x coordinate and b is the y coordinate.

Now for this reason a plane is called two dimensional. To locate a point in space, so in three dimensional space, three numbers are required. And we represent any point in space by an ordered triplet or order triple a b and c of real numbers. In order to represent points in space we first choose a fixed point call this o the origin and three directed lines through o that are perpendicular to each other called the coordinate axis and labeled the x axis y axis and z axis but usually we think of the y the x and y axis as being horizontal and the zed axis as being vertical and and we draw the or the draw the orientation at origination So we draw the orientation of the axis as in the figure. So here's how it typically is shown.

So there's the origin at the center there. And there's the y and x as the flat plane over there. And there's the z as the vertical over there.

So these are the coordinate axes. And the direction of the z axis is determined by the right hand rule. And as illustrated below. So there's your right hand here. Notice the thumb sticks out.

And typically you could also have the index finger stick out for the x axis. Notice it curls around here from the X to the Y and there's the vertical is your thumb. So if you curl the fingers of your right hand around the z-axis in the direction of a 90 degree counterclockwise rotation from the positive x-axis, from here to here is going to be 90 degrees like that. So this is a right angle and this is in other words 90 degrees like that to the positive y-axis. Then your thumb points in the positive y-axis.

positive direction of the z axis so these are all positive axes like that uh the three uh three coordinate axis uh yeah the three coordinate axes or axis determine the three coordinate planes illustrated in the figure a below so here's the coordinate planes so this uh x y we'll call this the x y plane over here and there's the y z plane is vertical over there and on this side you'll have the x z plane on this side here so those are the planes there and uh here's added more text uh this is also for my calculus book i just forgot to add it but anyways here so the x y plane so this is the x y right here is the plane that contains the x and y axis so there's the x and y axes and the y z plane contains the y and z axis so this is z there's the y and the x y plane i mean you actually know not the x y the x z plane x z plane contains the x and z axes and these three coordinate planes divide space into eight parts so you have one here then you have another one in the back and here uh three four and then you also have uh on the negative sides so you're gonna have another five six uh seven and behind that's gonna be eight so you have eight parts total and those are called octants and the first octant in the foreground that's this right here this is determined from the positive axis those are positive ones over there and uh and note here so here we have a room there's a left wall right wall and there's a floor and that's because well many people have some difficulty visualizing the diagrams of three-dimensional figures so you may find it helpful to do the following and you can see this figure be above so look at the bottom corner of a room so let's say here call that and yeah and call the corner of the origin and then the wall on your left is in the x z plane so there's the wall there there's left wall and the wall on your right is the y z plane so there's the right wall and the floor floor is the x x y plane yeah and the x axis runs along the intersection of the floor and the left wall so there's the x axis and then the y is on the right wall there on the floor and the y-axis runs up from the floor toward the ceiling yeah yeah this is the oops actually no that's the z-axis it runs up from the floor toward the ceiling along the intersection of the two uh of the two walls the y-axis runs along the intersection of the floor and the right wall so there's the y-axis floor and right wall the z is in between these left and right wall goes up that like that yeah along the intersection of the two walls and you are situated in the first octant so you can you're inside this room right here the first one and you'll know that there's going to be well eight rooms total and you can now imagine seven other rooms situated in the other seven octants three on the same floor and four on the uh four on the floor below all connected by the common corner point here so if you're in this room there's gonna be another one here and then three and then over here and then at the bottom you'll have four more underneath you all right all right so now if p capital p is any point in space let a be the directed distance uh from the y z plane and yeah we have from the y is that plane to p and let b be the distance from the x z plane to p and uh this word directed i think the calculus just means uh perpendicular uh to it or the shortest distance from that plane and let's see be the distance from the x y plane to p you know and we'll illustrate this all soon and then yeah we present the point p by the order triplet a b c of real numbers and we call a b and c the coordinates of p and a is the x coordinate b is the y coordinate and c is the z coordinate thus to locate the point a b c we can start at the origin o and move a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the z-axis as in the figure. below so we'll put all these words into a drawing to make sense of it so you'll have this z axis like that the vertical one now you're going to have our y axis positive one here and this is the origin zero and there is our uh this is going to be perpendicular to it this is our x axis like that and let's say we have a point p in space here so it's in 3d we'll call this point p a b c all right so let's go make a sense of what this says so a is directed distance from the y z plane so if this is the y z plane is this set up across on this wall so it's a distance from that uh and that's again that's just our x coordinate in other words that distance is going to be up to here somewhere or uh yeah so it's going to be up to here but before that so let's say we have this i'm going to draw this all the way down here all the way to the bottom and then um yes let's say it hits the bottom here and then draw this perpendicular it's going to go all the way across here like that oops and this is parallel to this one right here so this is parallel to this wall and this one here is parallel to this like that yeah so then this distance from the z y plane is the x coordinate and in other words we can start from the origin zero and move a units along the x axis so there's the a units and then b units parallel to the y axis that's b like that and then c units parallel to the z axis like this so b is going to be this distance b is it is the distance from this z x plane or the x z plane x z plane is b and then c is the distance from the x y plane to p so is the x y plane and there's a z and this is c not z like that and see this is all the way up to there all right so going further so the point a b c determines a rectangular box as in the figure below so so we have this right here let's draw this again but now you can see that that this can form a rectangular i mean a rectangular box so we'll do that again so we have the zed like that and then we're going to have our y i'm just going to make this yeah it's a bit shorter so this is our y and then this is going to be our x like this oops this is going to be our x like that and let's say we have our point p across here so this is our point p and we're going to have a box so let's draw this all in 3d this goes down here this goes draw this a bit neater so this is let's say it goes here and there is the line across there's the line across and then draw this all in 3d like that. And then this goes across like that. This goes like that, there is our 3d box.

Alright, so there's our box. And this one, we're going to call this P, A, B, C. Alright, continuing further. So if we drop the vertical perpendicular line from P to the x y plants, if we drop this line, just remove that in other words, our vertical is in C becomes zero, we get a point q with coordinates a, b and zero called the projection of P onto the x y plane. So that's this point right here, this is our point q like that.

And we'll call this a, b and then the vertical distance is zero like that. And similarly, r zero b, c and r the points are a zero a a 0 c are the projections of p onto the y z plane and x z plane respectively so in other words this one we're going to remove the uh this a value that's this uh horizontal x coordinate from here to here so in other words we're going to get this point here so actually no this one here we're removing it so we get this point here this one's going to be our r and then the x coordinate to zero everything else is the same b and c like that and that's projection onto the z uh onto the y z plane and that's right here y z plane and then the next one is our s we're going to drop this down yeah we're going to drop the y coordinate so we're going to project it all the way onto yeah onto this uh x z plane so the x z plane so we're going to drop this line and we're going to go all the way up to here so this is p q r and this one is Now let's put this like this. This is going to be our s. This is going to be our a. 0 is the y-coordinate, and then this is going to be c.

Like that. all right continuing further so as numerical illustrations uh the points right here negative four three negative five and three negative two negative six are plotted in the figures below so let's graph this one or let's plot that point out first so this one is notice it's going to go uh negative four so we're going to go the negative x axis then it's going to be positive three and then this is negative so in other words it's gonna be on it's going to be on the below these uh axis this is the z it's going to be a negative one so let's say we have our y like this and we're going to have to extend this all the way down because it's going to be negative z just extend this down further and we're also going negative uh negative x so that's going to go backwards so we're going to have to go from here this is our x but then we're gonna have to go to the backwards side of it so we're going to go negative four so let's uh put it all the way here so we're going to go negative four and let's put that line here and then we're going to go uh it's supposed to be all the way up to here and then we're going to go uh plus three to the y make it a bit shorter than that so let's make it um here let's go uh okay this should be good all right so this is good enough so there's three right there and then we're going to go all the way down uh we're gonna go all the way down to negative five so let's say the point is all the way across here we're gonna be drawing in 3d this goes it's going to go behind it so i'm going to break that apart and yeah here so this is going to be negative 5 all the way down and there's our point negative 4 3 and negative 5. and now let's draw this in 3d yeah so to do that let's draw our top part here and draw a parallel line like that and this is going to go all the way down and this more down like that until this is parallel like that and then we're going to have this line across here parallel to that a y-axis and then behind it we're looking to have dashed lines dash dash dash dash dash dash and then this is going to be another dash line like that and then dash line like that all right here so i fixed that up and made it smaller just so we can uh fit this inside it will fit the other one inside so this one is going to be 3 negative 2 and negative 6. so it's also going to be below this uh it's going to be below this floor because this negative vertical it's going to be yeah it's going to be negative six there but now it's gonna be in the front and then we're gonna go backwards there so let's draw this out so we're gonna do again the zed axis but it's going to be on the negative y so we're going to have a short z we're going to have a short y like that and then it's going to be on the positive x so we can make this super long there's the x like that and but it's going to be on the negative y so make it like that so negative y so we're going to go to 3 here let's say all the way to 3 here and then we're going to go negative 2 we're going to go all the way across here negative two and then we're gonna go all the way back down all the way until negative six so we're gonna go all the way down and we're gonna go like this negative six like that and then draw this in 3d and well this is the point right there and this is going to be our uh three negative two negative six like that and then we're gonna draw this all in let's make this bigger and now we're gonna draw in 3d so it's gonna go like that So it's going to go all the way down until it's parallel. Well, this one here as well until it's parallel.

So all the way down like that. And yeah, this one across like that. So make this all the way down like that.

Yes, it looks something like that. Now we draw this in 3D behind it. So this is going to be all the way down. Dot, dot, dot, dot, dot.

And we'll be down like that somewhere. And this is going to be dot, dot, dot, dot, and then and yeah here i just fixed it up made a bit smaller to match this and yes it looks like that in our rectangular box there oh anyways let's continue for this so the following cartesian uh product is the set of all order triples of real numbers and is denoted by this r to power of three and this is the latin capital r usually written like that which i sometimes just use r cubed there and this cartesian product this is just written as R, Osbertus in the Latin form. R X R or times R times R and this equals to the set related like this of all these points x y z where x y and z are elements of real numbers in other words these are all just real numbers so this is written like that so this is the set where x y z these are all just real numbers or it's a collection of all the points and it's for completeness so this r times r times r this is just indicating this is our r cubed equals all this all right so now that we have this this is just a set of all of these points x y z are of all these ordered triples and we have given a one to one correspondent correspondence between points p in space and ordered triples a b c in r3 where these are just all real numbers inside here so each point has a one-to-one correspondence with uh the uh yeah with with these collection of triples right here this x y z and so on for this a b c a b c and r cube so this is called a dimensional rectangular coordinate system and notice that in terms of coordinates the first octant can be described as the set of points whose coordinates are all positive remember yeah that's where all the positive axes are uh in two-dimensional and analytic geometry the graph of an equation involving x and y is a curve in r squared yeah curve or just a line like that's just a curve but in three-dimensional analytic geometry an equation is x an equation in x y and z represents a surface so in 2d it looks like that in 3d it's going to look like yeah just an actual surface imagine this is in 3d so a full surface like that and we can illustrate this in example one so example one what surfaces in r3 are in this set of all real triples right here are represented by the following equation a which is z equals to three and b which is y equals to five all right so let's take a look at the solution to part a this is z equals to three so the equation z equals to three represents the set it's all the points x y z where z is equal to three and which is a set of all points in r3 whose uh z coordinate is three and this is just a horizontal plane that is parallel to the xy plane and three units above it as in the figure below so the equation z equals three in three dimensions that's where all the coordinates are have a z equals to three variable or a value so if we have this let's just draw our uh x our z axis like that this is going to be our y axis or let's make it a bit more 3d so this is our y and then this is going to be our z i mean our x like that and this is our origin at the zero here this is our origin and now we're going to go all the points that are z is 3. so in other words we have to go all the way up to here three so here's a three point but it's a plane number so it's going to go to infinity everywhere so it's gonna let's just draw this in 3d so it's going to look like this and all the way across like that and uh since this is in front of it i'm just going to remove this remove this vertical i'm going to put this like this so this is above and now it's going to go below dot dot dot dot dot dot better so that's at the three this is going to be dot dot dot so that's the axis is behind it so we're drawing drawing this in 3d so it looks like that there and we could write here as this is z equals to three and this is a plane in three dimensions or r3 like that so in r3 So yeah, typically when I'm writing this by pen, I'll write it like this in Latin form, but when I type it out, I typically write it like this because it's easier, because there's no button on my keyboard for the Latin version.

Alright, so now let's take a look at the solution to B. So B was this equation, y equals to 5. So if z is equal to 3 as a horizontal plane, then y equals to 5 as well, it's going to be a vertical plane. So the equation y equals 5 represents the set of all points in r cubed whose y-coordinate is 5. This is the vertical plane that is parallel to the xz plane and 5 units to the right of it as in the figure below.

So if we draw our... this is our Z axis and we're gonna go across here, so that's our Z, I'm gonna put it something like this, there's our Y, alright let's make it even further, this is our Y and this is gonna be our X like that and this is where Y equals to 5, so let's go all the way up to here 5 and this is going to be a plane, so we're gonna draw a line like this line all the way down it's going to be behind it draw this in 3d just to make a just a giant plane like this like that and yeah as before here just fix it oh well made it like this and as before we got to draw the dash lines for the ones behind it i'm going to erase this right here draw this across there and then it's going to be dash line so this one's going to be straight across and it's going to be dash dash dash like that yeah so we have that line like there and also we could shade this in like that shade it in so it's a plane and shade this in like that and yeah so we have this equation is y equals to five and this is a plane in so a plane in r3 let's try this better All right, so now that we have that, that's a vertical plane right there. But now let's take a look at just a quick note on context.

Because this one here, we're asked to solve equation y equals 5 in R3. But if we're looking at two dimensions, so when an equation is given, we must understand from the context where it represents a curve in R2 or a surface in R3. So in example one, y equals 5 represents a plane in R3.

But of course, course y equals 5 also represents a line in r2 if we are dealing with two-dimensional analytic geometry as in the figure below so if we have uh there yeah this is just our x y axis this is our x this is our y then the line if this is our origin zero then the line is just like this this is just y equals uh this is the 5 in other words this is y equals 5 and this is a line you and r2. Alright, so now let's continue further. Now this section is out of the title here.

So equations of vertical or horizontal planes. So in general, if k is a constant, then x equals k represents a plane parallel to the y z plane. Yes, in other words, where x is fixed, so that's going to be a vertical Yeah, this one's gonna be a vertical uh plane and then y equals k is a plane parallel to the x z plane that's also vertical exist as a z then z equals to k is a plane parallel to the x y plane and that's gonna be horizontal and so on so in the earlier uh rectangular box figure above actually before we get that i'm just gonna write over here represents a vertical plane a vertical plane and this one's uh whatever z is constant horizontal plane parallel to the xy so in the earlier rectangular box figure the faces of the rectangular box are formed by the three coordinate planes x equals zero this is the y z plane y equals to zero this is the x z plane and z equals zero is the x y plane and also these right here x equals a y equals b and c z equals c and we can see this if we just scroll up to that earlier a rectangular box we had So that's this part right here. And what I'm going to do is I'm going to add all the points here.

So this is B, if we remove this A, it gets to this point, we'll just call the zero, B zero. And then this point across here, this is our zero, I know that's going to be actually this is going to be a zero. So we're going to remove this B there. So it's gonna be zero, zero, C, and also put this into the origin. and then we have this part right here yeah this part here is going to be our this is when we take this a right here and then but then we just remove the b is zero so we're going to get over here it's going to be a zero zero like that all right so here i just teleported above back down here and i just just added here so that same figure went over copied below so we copied here made it a bit smaller and now we could look at these six planes there.

And this top part is where the Z is constant. actually before we get to those let's go these this X equals 0 Y equals 0 and Z equals 0 these are the yeah those are the three coordinate planes here let's move it over here so we have the three coordinate planes and then we have also these extra planes so if we look at the coordinate plane I'm going to draw this in red where we have X equals to 0 let's see why is that plane Y Z that's in the background here let's make this a out in front of it so like this like that yeah and then etc so that is uh this part right here this is our x equals to zero like that and then the next one is let's make a different line here so this is this where y equals zero that's this can be x z plane this is y equals zero like that that's this or y equals zero like that and then this is a z equals zero is at the bottom where yet z is completely flat there the bottom like that and so on that's our z equals zero and then let's draw a different color it's a bit hard to see but just to illustrate and now this is the x equals a is when you have it all the way across this this face this is our x equals to a and then uh y equals to b is all the way across this face here and uh yeah x equals to b like that yeah so it's a bit uh messy there then we have z equals to c that's a that's a the top of this box but again z equals to c like that All right, yeah, so it's a bit messy there, but I think it's, if you were watching the video, you could follow along, and yeah, just follow along how I did, but if you're just looking at the notes, it's going to be a bit more complicated, but anyways, it just looks a bit messy, but I think it's good enough. All right, so now let's continue further, and I'll look at example two, and this one has two parts, part A and B, and part A states which points X, Y, and Z satisfies the equations.

x squared plus y squared equals 1 and z equals to 3 and then part b says what does the equation x squared plus y squared equal represent as a surface in r3 all right so let's take a look at the solution to part a first so because z equals to 3 the points lie in the horizontal plane z equals to 3 from example 1a so if we recall example 1a that this is our z equals 3. so it's going to be somewhere along this there but then we're also given an equation of a circle this is just equation of a circle so because x squared plus y squared equals one the points lie in the circle with radius one and the center of uh yeah and the center on the z axis as shown in the figure below so we draw this out we're going to have something that looks like this this is our z and there's our y axis our x axis there's the origin and this is our point three and and now we have it like that i'll make it a bit shorter then we have a point three here and now we're now so that's a three on the z but now we have x squared plus y squared that's the circle on this uh z equals three plane so we can write it like this right like this but there's a circle and uh yeah yeah that's just a circle there and well it's going to be behind this so let's make this like this dash let's try this one more time all right yeah so just have it like that so this is going to be our circle yeah so it's going to be circle and the equations are circle x squared plus y squared equals to 1 and z is equal to 3. all right so now let's take a look at the solution to b so b is here so we're given what does the equation x squared plus y squared equals 1 represent as a surface in r3 and in other words in three dimensions and this is the same thing as this but there's no z equals three so let's take a look at the solution to b so given that x squared plus y squared equals one with no restrictions on z we see that the point x y z could lie on a on a circle in any horizontal plane z equals to k so the surface x squared plus y squared equals one and r3 consists of all possible horizontal circles x squared plus y squared equals one and z equals k and is therefore is therefore the circular cylinder with radius one whose whose axis is the z axis so basically we're not limited to the z equals three it could be on anywhere across i'm just going to copy and paste that same graph here yeah so i'm going to copy and paste that and now we can remove this three so it does not have to be the three and it could be anywhere and this is going to go all the way down yeah we'll just go down here it's down here i'm just gonna i'm just gonna make it on the uh the positive axis but it can go all the way down and this is gonna be a cylinder like that a bit better like this and and then this is gonna be in 3d so i'm gonna erase this put this down here and then dot dot dot all the way down i'll actually keep going because it's a it's a surface not a 3d shape so i mean not a solid so it's not a solid so then put all the line all the way down there and go dash dash dash same thing with this dash dash dash dash dash extend it out just to make it 3d dash dash dash dash dash dash and then like that and i'm just going to shade this in blue just make it like this or actually in red shaded shaded cross or it's a bit hard to shade shade shade and then the inside shaded as well like that yeah just do that just to show it's a surface and then basically we have surface uh this is the cylinder or this is a cylinder right here all right so those are cylinders just typed it a bit better so x squared plus y squared equals to one and uh yeah so the radius here yes to the radius across here is one, the articles one like that, or instead of making it this way, this right articles to one here, articles one, yeah, from there to here. So these are going to be one, one, so. All right, so now let's continue further. And now let's look at example three.

And this one states, describe and sketch the surface in R3 represented by the equation y equals x. So let's take a look at the solution. So the equation represents the set of all points in R3 whose x and y coordinates are equal. That is, if we write this set out, we'll write it like this.

So this bracket x, and then x. y is x, and then z like that. And again, this is because y is equal to x. So we'll write them both as x.

And this is where x and z. you could write it as x z are all real numbers or you could even write it as x is an element of all real numbers like this and z is an element of all real numbers like that just calculus book wrote this method or in this form but you could write it as x and z as all real numbers or x uh all real numbers and z all real numbers like this and then just close bracket like that Now, this is just a vertical plane that intersects the x, y plane in the line y equals x and z equals to zero. And the portion of this plane that lies in the first octant is sketched in the figure below. And because it's a diagonal line, I'm just going to draw this out like this.

The calculus book draws it. So it has the z-axis like this. Here's the z-axis. So that's the z-axis there. And then it has the...

x-axis over at this side let's fix it up so x-axis over there and then it has the y-axis over like this y and then I'll put the origin here and that's just so that we could have this line better presented so I'm going to cross this line here so this is at the y equals x line and that z equals zero and then now we could draw a vertical line across I'm just going to make it shorter and then it's going to be like this it's shorter like that it's going to look something like this make it vertical and parallel and this is going to be like that and again because this is in 3d let's just erase this or we could do just move it up like this actually it's better just delete it and then start fresh so it's going to be like this this is going to go across like that this will be dash dash dash dash dash dash dash like that and then again if we just shade this out in red like that so that's the plane like that and this is at the y equals x in other words at the perfect center there so the same distance across and we're going to write this as a plane uh the yeah the plane y equals to x like that all right so now let's continue further now let's look at uh distance uh formulas well first let's look at the distance formula in 2d and then we're going to extend it to 3d so recall the 2d distance formula between two points and basically what we have is like this so let's say we have y x y axis just to derive it here and then we have a point here this is point one p one and this is our x one yeah call this i'll call this p one x one and y one then let's say we have a point here p two this is our x two y two like that So the distance between this and this, call this absolute value p1, p2, like that. So that's a distance. And then we could use Pythagoras, draw a right angle triangle. And then this distance here is going to be absolute value of y2 minus y1.

This is absolute value of x2 minus x1. And then Pythagoras. Yeah, and here I just typed it out here or just wrote it out. So then we just apply Pythagoras or Pythagorean theorem. And then square root.

we're going to get p1 p2 is equal to the square root and we'll only look at the positive square root is going to be square root over this one here let's put that for completeness well yeah if this is absolute value squared it's the same thing as writing x squared minus x1 squared plus y squared minus minus y1 squared like that whether it's absolute value or not when you square it's going to be positive it's going to always be positive so let's just uh just recall this formula right there all right so now that we have this and now the familiar formula for the distance between two points in a plane this one above here is easily extended to the following three-dimensional formula and here is the distance formula in three dimensions yeah so we have the distance p1 p2 between the points p1 x1 y1 z1 and p2 you x2, y2, z2, this can be extended over and it's going to be just p1, the absolute value of p1, p2 is the distance, this equals to square root and this just extends it over x squared minus x1 squared plus x, I mean plus y now, y2 minus y1 squared plus z2 minus z1 squared. Alright, so now that we have this, and then we're going to go over a proof of it. So proof to see why this formula is true.

So here, we just extended it and added the Z component here. See why it's true, we construct a rectangular box as in the figure below, where p one and p two are opposite of vertices, and the faces of the box are parallel to the coordinate planes. So let's draw this all out. So if we have a z axis like this, and there's a y axis like that and this is our z like this and then this is our x like this like that and let's draw this uh just in the top here just to get it high up so it doesn't interfere with the axes so let's say this is our point um this is our point p2 and let's say we have a point across uh here well i'm just going to draw a box first so let's say we have a box like this and just move it even here so the box like this and then this goes down like this this goes like that so we have a box and this is going to be parallel parallel until we have it vertical like this and then we have this across like that and then behind it, all the way to the bottom, this is going to be all the way behind it. All right, so we have this, make this a bit neater, or downwards more.

All right, and at this point here, we'll call this P1. This is going to be P1, P1, and this is going to be our x1 actually before we get that i just move this over here so we have some more space we're going to have p1 this is going to be our x1 y1 z z1 like that and now this point right here we'll call this p2 like that and erase this and draw a bit bigger okay so this is p2 p2 is our x2, y2, z2, like that. Alright so now we have this and what I'm going to do is I'm going to consider this point here as B.

This is our B point and this is going to be as x2, so all we're going to do is drop this z all the way down until it matches the z1 there. y2 and then z1 like that and this is I'm going to draw a line across here, dash line across and this is a perpendicular line like that. I'll just fix that up it's going to be like this perpendicular line and and now we're going to need this point across here from here to here this is the point that we want so this is um i'm going to write this as i'm going to write this as a distance this is going to be p i'm going to write this in red so all the distances i'm going to write in red p1 p2 absolute value all right so this point is b and now this point here is our a and i'm going to write this a as a is x two so basically it's going to be the same thing at this point here but we're going to remove this yeah it's going to be same as this one but we're going to remove this x1 and replace it with x2 so it goes over to here this is x2 this is going to be x2 and then everything else the same this is going to be y1 z1 like that all right so now we have this a b like that I'm going to write this as absolute value of a b like that and again this is uh i'm going to write this here this is another right angle so we have two right angles across here and there all right so we have a b there now we have this part here from the top to bottom this is uh we'll call this as p2 b like that and the next one here is from here this a to p1 at the bottom there that is going to be our absolute value of p1 and then a and these distances this is just well this is a horizontal distance remember it's parallel to this and and that just means well it's going to be x2 minus x1 the absolute value of it so we could write this all down so you could write absolute value of p1 a so this distance is going to equal to absolute value of x2 minus x1 and then this one right here ab this is parallel to this y axis so this distance is going to be the y's so then we have a absolute value or the distance a b is equal to absolute value of y y1 y2 minus y1 and then the last one is this vertical component it's just vertical parallel with this and so that's just going to be z2 minus z1 and that's our p2b like that so we'll write this as p2b is equal to z squared or I'm z2 minus z1 absolute value all right so now that we have this set up well we notice what we have is well we have two right angle triangles so we can basically find this distance which is our p1b and then we can solve this distance up top which is p1 p2 so we have two triangles there and I'll just change this back to red so we have this distance right here p1 and then b like that so p1b is down here so that's the hypotenuse of this triangle and then it will become the base of this other triangle that's the hypotenuse so we can apply pythagorean theorem twice so because triangles p1 b p2 so p1 b p2 and p1 a b so p uh yeah p1 a b right here our right angle two applications of the pythagorean theorem give and yeah let's just write it out so let's write our distance p1 uh b2 oops this raises so p1 b2 there so we're going to have uh p so absolute value p1 uh b2 and i'm going to square it just because pythagorean theorem is squared is equal to p1 b plus p2 b squared there so this equals two uh p1 b squared plus p um this is going to be uh p2 b squared like that so that's this part right here this and this is this over here that's what we have there and then then then the second triangle is this is the hypotenuse and this hypotenuse square is equal to this squared and this squared so in other words this part right here is going to be you p1 b squared is equal to p1 a squared plus a b squared and again this is just this part so in other words we can combine we could throw this inside so throw this inside here so that's what we can put inside and when we do that we get here so this becomes p1 p1 p2 squared is equal to and then we just replace that replace this with this p1 a squared plus absolute value a b squared plus the distance p2 let's play this better p2 b squared and we know what these all equal those are just these p1 a1 is x squared minus y minus x1 i'm an x1 i'm an x2 minus x1 not x squared and then a b is here y2 minus my one and then a p2b is this one here z 2 minus that one and that's all squared so we could write this as equaling to absolute value x squared minus x1 squared plus y2 minus my one absolute value plus z2 minus z1 squared and since we're squaring it's going to be positive so we could put this all together so thus and then and also square square rooted just to get it all out of the way and look at only the positive of the square root of p1 p2 is equal to square root and write this as x and then just write it in normal bracket x2 x1 squared plus y2 minus y1 squared plus z2 minus z1 squared and we can just box this all out like that so there is our distance formula in 3d All right, so let's just quickly double check. And if we scroll up here, yeah, so this is the one we were given. This is, yeah, the distance is just x squared minus x1 squared, and then plus the y's and z's, and then all squared.

So exactly the same thing as this right here. Now let's take a look at example four. And this one doesn't really give a question, just because it's pretty straightforward.

So the distance from the point P to, yeah, from the point P, which is two, the coordinates are 2, negative 1, 7. to the point q which is 1 negative 3 and 5 is so we just apply uh this formula there so pq is equal to square root and now the x x 2 minus x 1 so that's going to be 2 minus 1 2 minus 1 squared then plus y the 2 minus 1 so we'll just have this negative 1 minus 3 We're actually just for completeness because this is x2 minus x1 just to make it the same form. It doesn't matter because you're squaring it and it's always going to be the same. It's always going to be positive.

So let's just go 1 minus 2. Let's go from this as our x2 or our second point p2. So then it's going to be negative 3 minus 1. Negative 3 minus 1, that's going to be, I mean minus negative 1 is going to be plus 1. And then plus the next one is 5 minus 7. 5 minus 7 squared and now this equals 2 and well square root 1 minus 2 is negative 1 but you're squaring it and it's always gonna be positive just go 1 squared plus this one here negative 3 plus 2 or I'll just for completeness I'm just going to keep this in negative 1 squared plus and the negative 3 plus 1 is negative 2 squared and then plus 5 minus 7 negative 2 squared this equals 2 square root and then this is going to be 1 negative 1 squared is 1 negative 2 squared is plus 4 and then the next one negative 2 squared is plus 4 again that's going to be 8 plus 1 is 9 equals to square root 9 equals to 3 yeah 3 or or you could even have a negative 3 so negative 3 times negative 3 is also put that for completeness plus minus but it's just going to be that we just look at the plus side of it yeah so just uh just erase this so just do a complete this because when you're square rooting it's uh you can have still negative three times negative three but you can't have a negative distance in this case because we're defining it as absolute value all right so now let's take a look at example five and this one states uh find an equation of a sphere with radius r and center h k a 1. actually not one it's actually l so kaku's book writes l lowercase but it's it's just uh for me i like using capital case i guess because it's easier it doesn't look like a one because this one here i write one like that sometimes so this is how it writes l lowercase and i'll just write it as capital l uh so find an equation of a sphere with radius r and center uh c which is coordinates h k and l so solution by definition a sphere is a set of all points p uh x y z whose distance from c is r and you can see the figure below so let's just draw this in 3d so sphere is here's the z axis like this and there is the y axis uh this is our y and there is our x and there's z like that and this is we have the center here you the center inside is let's say c c and this is h k l and then the distance across all the way to because we're inside a sphere to p of x y z like that it'd be like this and then it's going to be always the same it's going to be r like that and just draw this like that and I'm just going to draw this, shade this in red. So it's a 3D sphere, shaded in like that, something like that. So it's a sphere.

All right, so we have this sphere like that, and the distance from C is going to be always, yeah, it's always going to be equals to r. And here I just move this over to make it more in the center there like that. All right, so that's actually in the center now.

of the sphere so thus p is on the sphere if and only if the distance pc equals to r and then if we square both sides we have we're going to have pc squared equals to well r squared and this equals 2 and the number of distance formula is just uh we subtract the x's and the y's and the z's and so on and squared and because we already uh we're squaring that so we get rid of the square root this becomes x minus, so this is the x, this is h, this k is the y and so on, x minus h squared and then plus y minus k squared plus z minus l squared. Yeah so thus our equation of a sphere with radius r and center c hkl is just this, so we can box this r squared over like that, so there's the equation of a sphere. all right so now that we have this and uh and basically this result of example five is worth remembering so we'll call this equation of a sphere so an equation of a sphere with center c hkl and radius r is and we'll write r squared on the other side so we're just going to write this as x minus h like this squared plus y minus k squared plus and then z minus l squared equals to r squared like that all right so now uh in particular if the center uh is the origin uh o then an equation of the sphere is so this is uh the c is all going to be zero zero zero that just means that the equation is just going to be x squared plus y squared plus z squared equals two r squared again uh compare this with the equation of a circle which is just x squared plus y squared equals to r squared all right so now this brings us to example six and this states show that the following is the equation of a sphere and find its center and radius so we have this equation x squared plus y squared plus z squared plus 4x minus 6y plus 2z plus 6 equals to 0. so solution uh we could rewrite the given equation in in the form of an equation of a sphere if we complete squares the first recall completing squares recall yes if we recall right here x plus a squared yeah this equals two because that's just a square so this equals two well x plus a times x plus a so we're going to multiply x by x and then x plus a and then x by a then a times x then a times a in other words we have twice we have an a times x and then x times a so this equals to an x squared plus 2a and then the last one is a squared plus a squared so x times x x squared then uh x times a plus a times x is 2a yeah 2a x like that let's erase this and move it uh 2a x and then the last one a times a is a squared like that yeah so the the key thing is we have this 2a and then we have this a squared there and that's what this x plus a squared is becomes this yeah so thus we just need to write these x and these y's and the z's all in this form so we can complete the square and what we'll do is now so thus so we have this i'm just going to blank this out thus and let's just move over this x over here just for completeness let's write it all out uh this is x squared and then plus at 4x so we're going to write this as uh let's write it uh this is right here x squared plus 4x put a bracket just to separate it and then the next one is is this uh this y and there's a negative 6y so we have a plus y squared minus 6y and then the next one is uh this is going to be plus z squared and then plus 2z like that plus 2z and then this 6 we're going to move it over to the other side it's going to be negative 6 like that and now if we look at this completing the square so this part right here there's a 2a there so all these have a 2ax to 2ax but we need to add a squared so in this case here we have 2a equals to 4 so that means a is equal to 2 and a squared is equal to 4 this means yes this means a squared is equal to 4 i'll also remove that arrow so we have a squared is equal to four this way it's small and now this case here we have so this right here this is put an arrow here all right so go back to here so this negative six that means we have our two a is equal to negative six that means our a is equal to negative three and then our a squared is going to be well positive nine like that and the next one is over here our 2a is equal to 2 this means our a is equal to 1 and a square is also equal to 1 like that yeah so what this means is well we need to add an a square to each one and then also subtract one from each equation so that we don't change it and then move that that additional a squared over to the other side just so yeah let's put it all over there so what we'll get is so we get x squared plus 4x minus i mean yeah this is going to be plus a squared now a squared is 4. so that's our a squared forward like that next one is and we're going to subtract by 4 squared as well this next one's going to be y squared minus 6y then plus a squared which is 9 plus 9 we're going to subtract that as well and then plus the next one is z squared plus 2 z i'm just going to put this here a squared for completeness plus 2z and this is going to be plus a squared it's going to be 1. i got and then at the right side is negative 6 so we want all these extra stuff that we added to cancel so we just add these on this side as well it's going to be plus 4 and then minus uh yeah i mean yeah this is gonna be plus four plus nine because they cancel we make them the same sign because it's on the other side and they would cancel yes we're not changing anything and then the next one is plus one like that and let's just add these up so negative 6 plus 4 is negative 2 like that and then these two is 10 so then negative 2 plus 10 this is just 8. yeah all right so this becomes and now this is completing the square so x squared plus 4x plus 4 that just becomes x plus a squared where a is 2. so this becomes x plus 2 squared like that next one's going to be uh this next one's going to be y this one right here a is negative 3. so we're going to have a y oops and go back here so y minus 3 squared and then plus the next one is so we hear this square completed uh this a is just one so we get x plus z plus one like that z plus one squared and this all equals to 8 like that so this equals to 8 and remember this equals to R squared and this whole thing is just an equation of a sphere, equation of a sphere, all right, so thus, and then we're asked here, so we're asked to find its center and its radius, so this is r squared, so thus our center, I'll write thus, the center is equal to, and remember it's going to be the negative of these, so this is x plus 2, uh it's going to be the negative because if we scroll back up to our equation here negative h the center is h negative k center is k and so on so we do the negatives so center is going to be negative 2 plus 3 so it's right at 3 and then negative 1 like that and the radius is i'll write radius radius so make it a capital actually the center and then radius ray d is is going to be uh r squared i mean r is going to be well is equal to square root uh yes square root 8. so here we square root 8 and this equals 2 which simplified calculus books usually does that 4 times 2 so the 2 becomes and the 4 becomes 2. so 2 times square root 2 and we're looking only the positive of the square root so that is the radius and we could just box out the center and the radius all right so now let's take a look at the last part of this video and that's example seven and this one states what region in r3 is represented by the following inequalities and we have this one is less than or equal to x squared yeah x squared plus y squared plus z squared and is less than four And we're also given that z is less than or equal to 0. So let's take a look at the solution. So the solution, the inequalities can be written as, and basically we're going to take these two inequalities, these two, this is greater than 1 and less than 4, less than or equal to 4 and greater than or equal to 1, and write this, and then basically square root both sides.

So what we'll do is 1 square root less than or equal to square root x squared, plus y squared plus z squared and this is less than or equal to four i'm squared four this equals to two so what we'll do is uh this basically equals to just one less than or equal to two yeah less than equal to uh squared x squared plus y squared plus z squared in other words that's a distance less than or equal to two and we'll only look at the positive because we're looking at distance right here And yeah, these inequalities right here represent the points x, y, z, whose distance from the origin is at least one, that's greater than one, and it's going to be at most two. But we are also given that z is less than or equal to zero. So the points lie on or below the x, y plane.

Thus, the given inequalities represent the solid region that lies beneath or on the xy plane and between or on the uh the following spheres pause right over the spheres and we're going to write this as x squared plus y squared plus z squared equals to one and in between these two right here and x squared plus y squared and it's just equation of a sphere plus z squared and then the r squared uh this is going to be r squared is just well two squared plus four and uh yeah so now if we were to draw this out uh this is going to be let's draw it like this so this is our um let's say i'll draw our z axis like that and then our yeah our y axis so that's a z and we're going to draw our y axis going across negative and positive there's the y like that and now we're going to do the x axis across here and there's our x axis and now what we'll do is well it's going to be two spheres distance one this this is the origin and we're just going to have a one here i'm going to draw a sphere here with distance one or circle across this well actually because uh it's going to be behind that so i'm just going to write it like this behind it and be on these be like that. Alright, here, I just fixed it up made it like that. And the next one is going to be a distance to be like this.

and again it's behind that z-axis all right here i just fixed that up so it looks like that and then it's again it's a solid region so let's write this let's draw this out like that it's going to be underneath this and then it's going to be behind this like that and now let's shade it in shade it up top here so make it look like it's 3d and shade it in etc whoops or actually i'll just shade it in like that so it's shading it in and now here fix up the shading like that so so yeah that's uh what it has just made it look like so it's an indented inside and there is our uh yeah half of a uh half of a sphere with a hole inside but then it's cut off so it's half of that yeah essentially like a thick coconut here that's cut in half oh yeah coconut with a thick skin there and uh yeah pretty interesting stuff and that is all for today and yeah so hopefully you enjoyed I'm just going to scroll back up and yeah I'll be continuing on this series in future videos on this vectors and geometry and 3d geometry and so on anyways that is all for today thanks for watching and stay tuned for another math easy solution

Transcript for:Understanding Vector Spaces in 3D Geometry

Transcript for:
Understanding Vector Spaces in 3D Geometry