 So we're back to to multivariable calculus and before I begin the new topic I wanted to to add a couple of things to our discussion last Thursday about Formulas for the area in polar coordinates remember we discussed polar coordinates and One of the questions which arises here is one of the questions Which arises here is to find the area of The following kind of picture you have a curve Which is given by the equation r equals f of theta Where r and theta the two polar coordinates, which we introduced last time, right? and in this polar coordinates a natural Bounds for such a Fortune such an area would not be the vertical lines like x equal a x equal b the way You know we used to do it in rectangular coordinates That's the old picture now. It's more natural to to give bounds in terms of the angle theta So it's going to look like a sector like this Between some angle alpha and beta And so we will assume that alpha is less than equal to is less than beta and theta will be between alpha and beta and so we look at this Figure which is bounded by the lines theta equals alpha theta equals beta and and the curve This is the curve And the question is to find the area of this So I just briefly touched on this at the very end of last lecture And I just wanted to tell you how to get this formula which I wrote down Without Without explaining it last time So the idea is again to to break this to break this picture into small into small angles Which would be? Some within some angle delta theta and And say here where you'll have are and And just look at the area of this sector in the disc, right? So we approximate now our curve not by not by horizontal lines or line segments But we will now approximate our curve by a small segment of a circle because that's more natural That's a more natural object in the polar coordinate system Okay, so so the question then becomes what is the area of Of this of this little sector in the polar coordinate system the analogous picture would be like this So we would have some delta x and delta y Right and delta y and so the formula for the area would be just delta x times delta y right and Or more precisely in fact we don't look at it as a delta y But we just look at it as a y coordinate over over graphs So it would be more precisely y times delta x and that's what after applying the procedure of breaking into small pieces Give us the formula for the for the area in terms of an integral like this y dx In other words y delta x the elementary area gives rise to this integrand y dx So we have to approach this problem in exactly the same way But now instead of calculating this area right here We need to calculate the area of this wedge of this little wedge of this little sector And this actually not so difficult to do because see the point is That we know the area of the entire disc So if you have a disc of radius r We know the entire area What's the area of the disc? Pi times r squared right areas pi times r squared and Okay, so now suppose we were given say one quarter of the disc What's the area of the of the of this of this quarter of the of the disc? Well be one quarter of this right so that would be one quarter why one quarter because the angle enclosed here is pi over 2 and So one quarter is pi over 2 divided by 2 pi Which is the like the full angle of the disc so in other words if you want to calculate the area not of the entire disc but of of Part of the disc which is enclosed within the angle of pi over 2 you have to multiply this result pi r squared by the ratio of the angle which is enclosed To the total area of the disc so it would be pi over 2 divided by 2 pi Right so that would be one quarter of pi r squared Which of course we knew in the first place because clearly this is going to be one quarter because there are four pieces Which are exactly the same but let's suppose You are in a pizza restaurant and you are you have a piece like this With angle theta and they didn't cut it in a very uniform way, so you don't know exactly it's not pi over It's not pi over 4 it's not pi over 6 or pi over 3 whatever and you want to calculate what area How much pizza you got and then compared to how much your friends got say maybe they got more So they should they should share maybe to give you back Give you back some So what would be the answer? Well, we should we should argue in exactly the same way as before Before that angle theta was pi over 2 and what we did we took the ratio between pi over 2 and 2 pi And now instead we'll have a ratio of theta to 2 pi. That's That what that's what will represent the piece that we got as Opposed as a sort of the ratio of that piece to the entire disc to the entire pizza And therefore the area would be pi r squared times theta divided by 2 by 2 pi Right, so pi consoles out and you end up with one half r squared times theta That's the area of the sector of angle theta Which is exactly our question except in asking this question. We called it delta theta instead of theta so this area is Is one half our R squared delta theta and that's the analog of this formula y delta x So just like this formula y delta x gives rise to this integral This formula for the elementary area of the of the of the elementary wedge of the angle delta theta will give rise to the integral from alpha to beta one half r squared D theta you see it's exactly the same just kind of replace delta theta by d theta and That's how we get that formula which I wrote down at the end of last lecture So to summarize The key to calculating the areas like this is to be is to break the The figure that you got into small pieces Which will then look like elementary figures like in this case would be an elementary Rectangle, which is where I magnify a blow it up and I show it there Calculate its area and then it gives rise to the formula for the area of a general picture like this Whereas in the polar coordinate system you do exactly the same But you kind of follow the shape of the figure is different because the coordinates are now different So now we split it into little sectors like this where the angle is delta theta I magnified one of those pieces right there. I found the area of that little piece by thinking about pizza and Calculating it right and and then I put all these little pieces together and this gives gives rise to this integral Okay, any questions about this? So that's the way to calculate That's the way to calculate the area in polar coordinates and there is a similar formula also for the arc length in polar coordinates But the formula for the arc length I don't want to spend time on it because we really take the formula which we derived in the in the rectangular coordinates About a week ago, and we just make the substitution where x is equal to our cosine theta and y is equal to our sine theta And then just see what comes out what comes out is some in some formula in terms of r and theta Which you can then use to calculate the arc length So that's the way we do it for the arc length here It's more conceptual because we change the point of view slightly instead of looking at rectangles We look at these sectors It wouldn't be wise to try to break this into rectangles because they don't fit in this picture What fits perfectly in this picture are these little sectors So then we need to calculate the area of the little sector which we've done here Okay, so this Speaking of pizza analogy this were the leftovers from last lecture and now It's not it's not what you say. It's how you say it, you know, so delivery is So I'm working on it now So now we go we move on to our next topic And the next topic is vectors and geometry of space and space here means three-dimensional space Which surrounds us you see up to now in the last three lectures. We talked about objects which are defined on the plane Here I've been drawing curves on the plane and We've discussed various things about those curves like arc length or various areas which are enclosed by those curves But the basic point The most important aspect of what we've done up to now in this class is the fact that we have worked Exclusively with object confined to a plane and by a plane I mean this blackboard or sort of infinite an infinite extension of this blackboard in all directions Right. All our objects have been defined on this plane But now we want to move on as I told you in the very beginning. We are in this class We are going to study not just objects on the plane but also object in space in the three-dimensional space and In the three because we live in a three-dimensional space If you don't think that if you don't count the time So in fact we live in four-dimensional space time, but that's that would be topic for the next calculus class I guess four-dimensional space now will do here in this class We do two-dimensional space the plane and three-dimensional space So three-dimensional space means more interesting because it fits more objects more of now It fits not just curves and there are a lot of interesting curves Which are appear in space which cannot which are not confined in any particular plane But also perhaps more interestingly it contains objects of dimension higher namely two-dimensional objects the surfaces like a sphere for example Or or or this this this waste basket These are all surfaces if you if you ignore the fact that actually they have certain thickness So what we'd like to do ultimately in this class is to understand the geometry of those surfaces as well as curves and to be able to answer various questions about them like finding the areas and Even more even more sophisticated things So what we do what we need to do now is to sort of lay the groundwork for this in other words We have to develop some formalism because how are we going to represent curves? How are we going to represent surfaces in a three-dimensional space? What's the most efficient? so always check for batteries Before you begin so in space we need to we need to develop some technique in order to attack these problems and The very first thing that we should talk about is a coordinate system Because you know and on the plane it sort of goes without saying that we started out with Whenever I start writing drawing a diagram drawing a picture or drawing a curve I draw this coordinate system. We're kind of it's almost like a reflex I say and usually we call them x and y the two coordinates After last week we kind of we're wiser now We realize that this is actually not the only coordinate system that we have available for for the plane For instance the polar coordinates provides us a different coordinate system But this is the basic one and we should start thinking about a similar coordinate system in the three-dimensional space So this is the first step that we have to make coordinate systems Now of course the problem is that even though I'm going to talk about the three-dimensional space I'm still going to draw things on the two-dimensional blackboard, right? So of course I cannot draw this wastebasket on on this on this blackboard what I can draw is really a projection So I kind of we kind of create the illusion that there is some depth in the picture and we try to imagine that The three dimension the third dimension is there somewhere So the the way we know we usually would draw coordinates Would be like this so that of course the point is that they are not on the same blackboard only This one and this one lie on this blackboard and this one is sticking out He's sticking out by what we are looking at kind of slightly from above and And that's why it appears to us this way So now we want to we want to label this coordinates in a certain way in the on the two-dimensional plane we label them x and y and In fact, some of you may have wondered Why we label them x and y and not And not for y and x not in the opposite Not in the opposite order why not like this Of course in a sense it is the notation is arbitrary. We have chosen These letters once and for all so in a way you can say that it's not a meaningful question Because it depends on what we mean by x and y but of course What is important here is not so much which letters we use but the ordering the implicit ordering between x and y X comes first and y comes second because that's how they are ordered in the alphabet, right? So we could use y and z and again We it will be assumed that y goes first and z goes second so there is actually a basic asymmetry between these two coordinates x and y and We don't talk about it's much. We haven't talked about this much when we talked about curves but in fact now is a good time to To discuss this and to realize that there is a basic asymmetry the point is that You could work with a coordinate system like this xy Or you could work with a coordinate system yx and these are not equivalent to each other There is no way to transform one into the other Without removing them from this plane. Of course if I were able to Maybe I should draw it more like this if I were able to pick it from the plane and flip x and y I Would of course then I Will get back to it to this picture x and y that is a more traditional picture we draw right But we cannot flip them within the plane You can argue. Well, what if we just try to move this coordinates like the You know on the clock and kind of move them and make them go through one another But that's not allowed because then you they would have to pass through each other and at that point They will cease to be a coordinate system. They will become parallel to each other So that's not allowed. So in fact you think about you will see that there is no way to transform one into the other There's no way to transform one into the other by by moves confined to this plane and That's a very important point which tells us that actually our plane has two different orientations Choosing one or the other this coordinates this coordinate system or the other coordinate system Gives it orientation and the funny thing is suppose that there was another class behind that blackboard and There were people there like a mirror image of this one the bizarro world if you will and suppose there were kids, you know sitting there and also learning multivariable calculus then This one one of them The the first one would appear to them as the second and conversely I mean not exactly, but if you if you rotate that that's exactly what you'd get so What in other words the orientation from this side? There is orientation from this side when you look at this blackboard from this side And there is orientation of this blackboard from the other side from the bizarro world side and these are two different orientations There's a two different worlds and you can't transform one world into another Within the plane within the confines of the plane if you go into this three-dimensional space You can't do that you can just pick it up and flip them But inside the two-dimensional space you cannot do that at first it appears as a kind of a nuisance And you may think that it's not a very important point but actually And in fact for the plane it's not so important, but in three-dimensional space it becomes more important and that's why we emphasize it In three-dimensional space. We also have different choices of writing coordinates labeling of labeling coordinates and in three dimensions we will label coordinates as x y z and Of course again the most important point is not so much which letters we use But the ordering in which they appear and the order of course will be the natural one one two three Just the way they appear and the alphabet But then the question is how do you assign there are there are now many choices to assign these letters to to this coordinate axis and The question is are we going to get always the same coordinate system in other words are we going to get coordinate systems? Which can be transformed one into another by moves within the three-dimensional space? And so now since I've shown you that in two-dimensional space on the plane That's not true in other words on the plane There are two different coordinate systems, which cannot be transformed into one another you would not be surprised to know that actually Here as well in three-dimensional space. There are also in equivalent coordinate systems But you might be surprised to know that again there are only two choices not more than two So the choices are basically The one which we will always use will be like this X Y Z and the other choice Which we will not use which is not equivalent to this one which is which is a bizarre world coordinate system is this one In other words we switch X and Y just we did before on the plane Just the way we did before on the plane If you try the other choices for example, you put X Y Z like this There you will see that there are many different ways. There are six different ways to label them You will see that That picture can be transformed by simple rotation into one of these two pictures So really there are only two in equivalent ones up to rotation within the three-dimensional space If we could go into the four-dimensional Into four-dimensional space if there were a fourth dimension that we could visualize Just the way we can visualize the third dimension when talking about the plane We would actually be able to pick it up and transform one coordinate system into another in that four-dimensional space But since we don't have for its fourth dimension or more precisely the fourth dimension time is kind of elusive It's not easy to visualize it. So within the three-dimensional space. We cannot do that So we have to agree from the beginning on what is the rule What is the rule for labeling these coordinates because we have to do it now because from now on We have to use the same system. So we are on the same page or on the same blackboard now What is the rule and So the rule is there are different ways to phrase the rule and that the rule is is in the book It's explained by using fingers But I always forget that so I will tell you the rule which I use Which is you know, I grew up in Russia and We use the rule which is called the corkscrew rule And this is not to say that Russians like to drink But anyway, it is called the corkscrew rule and the way it works is like this If you rotate the corkscrew from x to y So here is the corkscrew I Kind of draw the most the basic one if you rotate it from x to y Then this the thing will go in the direction of the of the z of the z axis right in other words if you Well, it's not a good picture because then the bottle would have to be upside down, but you see what I mean so if you think about the corkscrew going from x to y which is natural right going from x to y is like going from the first coordinate to the second in Going from x going in making a rotation as if you were moving the x core x coordinate system to the y Sorry x axis to the y axis then the screw will move in the direction of the z axis So that's the rule which I find easy to remember Even if I don't have a corkscrew It's easy to remember, but you can use whatever rule you want You just have to remember that there is such a rule and to draw this picture in this way Okay, so that's the first point that we have to make about this the second point is we have to develop we have to Develop some tools for representing objects on on this on this in space Okay, so The first I already told you that the objects will be mostly mostly interested in Still looking for battery, but I guess So the objects that we are mostly interested in our curves and and surfaces But in fact there are simpler more elementary objects namely points So before we talk about you know, it's complicated matters like curves and surfaces. Let's talk about points In the case of a plane we know how to represent a point. We just drop the perpendicular lines onto the x and y axis right and Say we get x zero and y zero and we represent the point as x zero y zero like this And we are going to do the same In the three-dimensional space So I draw the same coordinate system and I'm thinking okay if I go from x to y they're going to be z So that's how I should put them Because if I put them in the other way, I would have to move the corkscrew this way and it would go down So z would have to go this way So this is a good orientation This is this is orientation. We've chosen and Our mirror image will have the opposite orientation. So if we have a point We are we also can represent it by with coordinates and now they're going to be three coordinates Obviously because we are in three-dimensional space and the way we find this coordinates is as follows We drop the perpendicular lines line from this point to the plane x and y Now you have to at this point You have to use your imagination and you really have to think of x and y as a plane which is sticking out like this And the point is actually not here. It's not a it's not in the z y plane, but it's here so you've got Like I can't show you because my one of my hands is busy with the microphone But you can you can I'm trying so this is a plane and the point is here So you drop the perpendicular down like this so you get a point on the plane Which is the x y plane now I draw it like this So the point is somewhere in space and that's what that's what that's where it drops So now I get a point in the x y plane But a point in the x y plane we already know how to represent We just drop the perpendicular lines to the x-axis and y-axis. So what we got already is y zero and x zero Coordinates of this of this point the two of the two of the three coordinates. What about the third coordinate? It is tempting to draw a perpendicular like this But that's not true right to give it really a depth to give it the illusion of a three-dimensional picture I have to do it in the following way. I have to connect the origin to this point, which is the Which is the point I got by dropping my original point on the x y plane and I have to Connect To the z-axis by a parallel line. So this one and this one are parallel to each other and that's going to be the z zero coordinate You see the point is that This is perpendicular to this entire plane. So in particular is perpendicular to this and that's why the z-axis is perpendicular to this line So what I'm doing really is I'm dropping perpendicular to each of the three coordinate systems But the way I draw it is slightly complicated because I'm using two-dimensional projection of the picture It would have been much easier if I had a three-dimensional model of this whole thing So then you would see more clearly what are the what the coordinates are, but I think it's fairly self-explanatory here so If we have this picture, then we will say that the point has coordinates x zero y zero and z zero And this is in complete analogy with The situation that we have the picture that we have in two dimensions. So nothing nothing surprising here What else do we need to know? We need to know the distance between two points Right, so for instance, we would like to find find often times Let me use a different chart often times we might might want to find the distance from From this point to the origin Let's call this so we need a formula for this distance and In two dimensions, we know that this distance is computed by a very simple formula namely square root of the Xzx squared plus y squared and now the formula is going to be very similar It's going to be the square root of x squared plus y squared plus z squared Okay so So very similar instead of sum of two squares like this you have the sum of three squares And then you take the square root and that's how you calculate the distance from this point to this point And it's very easy to prove this by using Pythagoras theorem. It's in the book. I'm not going to get into this It's very easy All right, what if we were What if we were to calculate The distance between two different points Actually So you could have a point P and then you have another point P prime and say here you have coordinates x 0 y 0 and Z 0 and here you have coordinates at x 1 y 1 and z 1 For this one so the distance so you see the point is that This you can find this distance you can just parallel transport this interval to the in such a way that the first point Would end up at the origin and if you do that the second point We'll have coordinates which are just the differences between the coordinates of the original two points. So at this point we'll have coordinates x 1 minus x 0 y 1 minus y 0 and z 1 minus z 0 Okay So the distance let's call it. Let's go D Here the distance will be then according to this formula the square root of x 1 minus x 0 squared plus y 1 minus y 0 squared plus z 1 minus z 0 squared just like this Is this clear to everybody? Okay, and and this already gives us a way to describe a very important surface on in space namely the sphere Because the sphere is defined as the set of all points Which are equidistant from a given point? so for example Let me raise something Let's talk about the sphere of radius R With the center at the origin These are all points whose distance all points P whose distance To the origin origin I denote by by this letter zero. This is not to I don't mean the The number zero, but I mean this point. It's really not zero, but it's point all looks like zero So points whose distance to the point all to the origin is equal to r and R is is some fixed number, right? So what does it mean? It means that the square root of x Square plus y squared plus z squared Where x y and z are coordinates of the point P? We'll have to be equal to R Because that's exactly the distance and if we say that the distance is equal to R that's that's the equation we get Now it's it's what it's it's a good idea to Square both parts both side left hand side and the right hand side So then we get the equation x squared plus y squared plus z squared is equal to R squared And that's remarkably similar to the equation of the circle. So compare with the circle Circle for which the equation is x squared plus y squared equals R squared Here now we have a third variable And so we just add the third square to the left hand side, but otherwise equation looks very similar so now we have an equation of the Out of the sphere with with this centered at the origin and again You see remember the rule which I explained to you at the very beginning that if you want to calculate the dimension of an object You have to take the dimension of the ambient space in this case 3 and subtract the number of equations here I've written one equation so the dimension drops by one and We get a two-dimensional object. So we get the sphere which is two-dimensional Two-dimensional object it is a surface Surface is by definition a two-dimensional object Unlike wise we can write the equation for a sphere which is centered not at the origin But at any given point by using this formula So if I say that this is equal to R that would be sphere of Radius R centered at the point P That would be the equation for x1 y1 and z1 with x0 y0 and z0 fixed being the coordinates of the point P alright So we already have some rudimentary knowledge of the three-dimensional space and we know And the equation of one basic basic surface in it So what's the next step? The next step is to introduce objects to to introduce a new tool to introduce objects of a different kind Which you have not used yet in this course. These objects are called vectors and you will see that these objects are very Very useful for Precisely for studying the kind of questions that we'll be interested in like finding areas Representing objects and so on so the next step is to is to talk about vectors. So how many of you know vectors? So maybe we should just go home Okay, but let me just go briefly it'll give you an overview. So I will not dwell on it too much then So what is a vector? Vector is an object which is which is different from numbers Which we have studied so far. So so far we studied different options. We studied we know numbers. We know points for example on the plane Points on the plane are not numbers in they are represented not by a single number by by two numbers Right and in fact, they are not They're more somehow than a two numbers because they are a gemat a point is a geometric object Which is positioned somewhere on the plane or in space? So a point is something which has a position which is a position on the plane or in space and A vector is an object of yet another kind, which is characterized by two properties It has a magnitude or length and it also has a direction So so we usually draw vectors as as as directed line segments So it is convenient to to draw a vector as The line segment like this connecting a point a to a point B And You see when I do that I give it two properties one is a direction Because this arrow points in that direction and a second is a magnitude or the length and the magnitude is the deal Is the distance between the point point a and point B? okay But now the question is how do we how are we going to represent vectors in practice in other words? In practice, we will represent points by their coordinates So we should discuss also how to present vectors by something like coordinates and various operation on vectors What kind of things can we do with vectors and? Here there is a subtle point, which is very important, which is that? Which is that in fact a Vector is not really a picture like this But vector is really what I said Something which which is just determined by its direction and and its magnitude in other words if you take And not if you take a parallel if you take a parallel transport of this if you just move It to another pointed Directed segment like this we say points a prime and B prime And you make sure that the directions are exactly the same in other words that these two lines are parallel And they point in the same direction and the lengths are the same Then actually we will not distinguish between these two vectors You see we will not distinguish between them So the vector is not really a pointed a pointed segment or directed segment It's a whole class of directed segments. A vector is this is one representative of of This vector. This is one possible way to represent our vector as a Directed segment going from the point a but you might as well start at the point a prime and you would represent exactly the same vector So usually we will denote vector By small light and letter like a and we'll put an arrow above it and In the first approximation you can really think of a vector as just an arrow like this Which has a directional magnitude, but you have to realize that many different air arrows represent the same vector Namely all parallel arrows Parallel to each other pointing in the same direction of the same length represent the same vector So then of course the question is how it can be possibly Work with these guys because they are there are so many possible ways to represent them Well, that's where we have to start using our coordinate system So we draw our coordinate system and again, we remember the rule. That's why see Now as I said the same vector a could be written could be represented many different ways, but To to find such a way we have to pick the initial point once we pick the initial point Then it's it's already the vector the the arrow is determined by that If we're drawing that vector a But amongst all the points on the plane once we introduce the coordinate system. There is the origin There is a special point so we might as well We might as well draw a representative from this class from this whole set of possible Arrows Representing this given vector we can draw one of them which starts at the point at the point of the at the origin of this three-dimensional coordinate system. So the result is And I'm trying to draw it parallel to this one in some sense But although in my previous discussion, I kind of I was kind of working on the plane But now I'm working in space So in fact this end point is not necessarily on the plane is not in necessarily part of the blackboard But it could be again hanging somewhere in the middle like on this picture Right So now the vector a gets a much more concrete realization It is a pointed Interval or directed interval the directed segment But now it goes from the point O from the origin to some point P So that that's already much less ambiguous In other words once you introduce a coordinate system, you have a preferred representative for all possible arrows giving you the same vector and Then what we'll do is we will write that a is equal to OP or in this case we'll write that a is equal to AB or a prime B prime In other words different notation in which you use the initial point and the end point and again put put the arrow on top So from this point of view It looks like a vector is essentially just determined by its end point Because we agree to use as the initial point the origin So the only freedom we have is where to put the end point So it looks at first glance it looks like there is no difference between a vector and the point But this is misleading you have to realize that this is misleading and actually there is a lot more to the vector Than there is to a point even though when we represent it in this way The vector will essentially be determined by its end point once you center it or make the initial point to be at the origin And really the main difference is in the meaning of this in the physical meaning physical interpretation of Of a vector as opposed to a point the point is just a point. It's located somewhere and It knows nothing about the rest of this of space But vector really is something else Here we talk about a vector being a pointed interval But in fact a much better way to think about it of a vector It's much better to think that the vector is a transformation of space a vector is a shift of space because What do you need to know if you want to shift something what do you need to know? well, it's it will be much easier to to express this in instead of Instead of a three-dimensional space it will be much easier to explain this on the plane So let's use this analogy So I have this table. I have this chair. Okay, so It's on the plane. So think of it as a kind of representation of a point on the plane where the plane is the floor or the podium really, okay So I want to shift it. So I move it in this way. What do I need to know? To what do I need to say in order to explain how to how to move it? In other words, if I ask one of you to move it I have to explain which direction and how far, right? So and suppose I ask you to shift everything in other words, I can ask some workers to come here and shift this podium I would say I want to be closer to you So I want to move this podium, you know Whatever one yard closer What does it mean? It means that each point of this podium will be moved by one yard in the same direction That's the vector a vector is a rule By which you shift all points and then if you look at the particular point like a point a Then a vector will displace it or shift it to a point B If you look at another point a prime this same vector this same rule will shift it to a point B prime And the point the origin will shift it to this point B So this is a much better way to think of a vector as a as a rule for displacement for sort of parallel Transport of the entire plane or the entire space If you have such a rule then you know where each point goes and that's what each arrow represents and now it becomes much more clear why different arrows as Long as they are parallel and have the same magnitude Correspond to the same vector because they are part of the same rule part of the same shift rule. You see what I mean Yeah Any questions about this this is an important point to to remember to kind of understand better what's The kind of stuff that we're going to do with vectors All right, but that's sort of a physical interpretation But now more more concretely we would like to We would like to work with vectors and try to represent them In much the same way as we represent points say So a point has three coordinates So now we want to to represent vectors also in a very concrete algebraic way What I talked about up to now is kind of geometry. I draw pictures and I explained the geometric mean But now I would like to do some algebra. I want to represent my vectors by by some coordinates or some algebraic objects and It's clear what we need to do We simply need to place the vector the initial point of the vector at the origin and we have to keep track of the endpoint and the endpoint we already learned how to How to Describe we describe it by its three coordinates X zero Y zero and Z zero right So if that's the case then we will write the following we will write that a Is equal to X zero Y zero and Z zero So what happened? We are using exactly the same information as the information provided by the point P But because now the point P is really not the central object, but it's something which is just We just sort of an auxiliary object which we just use to understand the vector because that's the endpoint For the pointed interval which we obtain when we apply our displacement at the origin, right? So the point P sort of has plays a secondary role here Nevertheless the point P has a very nice representation by X zero Y zero and Z zero So we might as well use this information because it uniquely determines our vector Because we agreed that the initial point is at the origin So all we need to know is the endpoint and the endpoint is just this so that's why this information is sufficient To represent a given vector, but when we write it like this We want to distinguish this notation from the notation Which I use for the point P itself for the point P itself. We use notation with the round brackets And we don't want to write it here like this because if we were to write like this We would be saying that this is a point, but it's not a point as I keep trying to explain to you It's not really a point the point in question here namely the point P is really just the endpoint of the vector But the vector has a lot more Carries a lot more information or has a different geometric interpretation than just a point So we have to separate the notation for the vector and the notation for the point and where we do it We use this angle angular brackets instead of the instead of the round brackets And to someone it may appear like we are too pedantic and it may appear like what's the difference? Well, the difference is if we will see now that the vectors actually have a lot of structure We can add two vectors to each other. We can multiply a vector by a scalar and so on a point A point doesn't have such a structure or points don't have such structures. We are not allowed to add two points We're not allowed to multiply a point by a number a point is just a point. It's static It just sits there a vector is a kind of a dynamic thing Which as like I said is a transformation of the space of shifting everything in the same direction and by the same magnitude So if you have two vectors two such shifts, you can actually do them You know one after another and therefore you get a third vector you get the sum of the two vectors and so on So in that sense a vector really is a different object on a point And that's why we really have to distinguish between vectors and points and use a different notation So this is the notation notation number one notation number two we Amongst all the vectors that we have on the plane we introduce the basic ones The basic ones are the vectors which go from the origin In the direction of one of the three axes I mean from the origin in case you want to apply it to the origin But in fact as I said you can you can apply it to any other point as well so this are There are these are the following three vectors one of them One of them goes along this axis and has length one It's uniquely determined by this because remember vector is uniquely determined by direction and the magnitude So amongst all the directions in space once we introduce coordinate system You have three basic directions x y and z so why not use those three vectors? But then you have to say which magnitude and the simplest magnitude would be one So you got yourself three different vectors this one this one And this one and we'll call them I J and K and so the point is that we can also write this vector as X zero times I Plus Y zero times J Plus Z zero times K in other words We can decompose any vector as a combination of multiples of the three basic vectors Which of course immediately begs the question as to what do I mean by? addition by the addition of vectors and what do I mean by multiplication of a vector by scalar, but I'm kind of Slightly jumping ahead because I know that you already know most of this material Let me just remind you Let me just remind you that a simple rule how to add two vectors There's a so-called parallelogram rule. Do you remember the parallelogram rule? Yes, or no, no Oh Someone said yeah, but most people don't remember should I remind you I remember Okay, so the rule is like this if you have two vectors a and b and You want to calculate what is a plus b? Then the way you do it is you apply the two vectors to the same point Again a vector is a rule. It's a displacement rule. So each point knows where it goes Right. So now we have two vectors. So we have two rules. So let's apply Let's apply both of them to the same point and it depends in your in your in the problem You're solving there will be some natural point to which you would want to apply apply them Now to find a plus b what you need to do is you need to Draw a parallel gram which is spanned by these two vectors. In other words, you draw a parallel line to this one Parallel to this vector and at this point you draw a parallel line to this vector So this and this are parallel and this and this are parallel and then these two lines will intersect somewhere And that's your that's the endpoint of the sum So that's the vector a plus b if this is a and this is b Then that's a plus b explanation of this rule Like I said, it's better to think of a vector as a displacement rule So now you've got yourself two displacement rules a and b Each of these rules can be applied to any point whatsoever. Okay, let's apply first. Let's apply first of them a and then apply b Consequently, let's apply them one after the other So first we apply a now we have to say to which point do we apply it? Well, let's apply it to this point There is the end result is that we Get here to this point after this I would like to apply the displacement rule B And I like I said this placement will be can be applied at any point here I drew the result of its application to this point What will be the result of its application to this point? Well, I have to take a parallel line of the same length This will be a directed interval which will represent the same vector, right? So I end up The net result of this two Two displacements this one and this one is that I end up at this point That's why the composition of the two or the sum of the two vectors is this vector So it's a very simple geometric Interpretation if you draw the picture using this parallel gram. This is called parallel gram rule if you draw it using this triangle is called a Triangle rule by the essence the the meaning is the same is this clear Okay We can also multiply we can also multiply vectors by by scalars When I say scalar, I just mean usual numbers real numbers What does it mean? Well here we should think we should think of a vector as a Direction and the magnitude when we multiply by scalar we we Well, it depends on what kind of scalar let me let me rephrase it If you multiply by a positive number if you multiply by a positive number This will be a vector in the same direction and the magnitude Will be c times the magnitude of the original vector So the picture will be something like this. This is the original vector. Let's say you want to take This is a and you want to a two a is going to be like this So this distance will be twice the distance of a Right, and you have the same direction If you want negative two or if you want any negative number You will have to be the opposite direction. So this is for example negative two a so Same direction see less than zero opposite direction opposite direction So you have two operations addition of two vectors and multiplication of a vector by scalar and And Here I combine these two operations. I take my vector i and my I am multiplied by x zero What do I get well according to the rule I have to get something which goes again along the x axis and now has length Which is equal to the to x zero times the length of this guy, but the length of this guy Just like the length of the second guy and the third guy J and K Is equal to one So when I multiply by x zero, I get something which has length x zero So it's just the vector which ends at this at this point And likewise if I take J times y zero I get a vector which ends at this point And if I take the third one K times Z zero I get this one and now it's not difficult to see That if you take the sum of this vector this vector and this vector you will get this one I'll leave it for you to figure this out So that's that's how you get the second the second representation of a vector This is the first one where we just keep track of the coordinates of the endpoint This is a little bit more descriptive because it really emphasizes the fact that it is The displacement is a superposition of three displacements one in the x direction one in the y direction and one in the z direction All right, so what's next? Next we have to Work out some tools for for dealing with vectors and And these two these tools are the so-called dot product and cross product So how many people know what dot product is? Okay, that's better So I'll go quickly over this By the way an important point is that I tried to do all of this in full generality in other words in a three-dimensional space So you see that I'm writing everything in space. I have three coordinates My vector has three components and so on but you can do the same analysis on the plane if you are on the plane if you are on the plane you can also Represent vectors in a very similar way the only thing that will be different is that you will be missing the last coordinate the last component That's all right because now you will have two coordinates x and y you can still talk about vectors on the plane and You can transport this vector to the in such a way that its initial point is the origin and then you will have two Coordinates we have some point P. You'll have two coordinates x zero y zero and so your vector Let's call it again a Sector a would be written as x zero y zero So see just to just two components instead of three So if you do a homework exercise and you are given a vector in this form That's how you know right away that it's a vector on the plane and not in space because you only have two components Or you can also write it as I x zero times I plus Y zero times J where one more time I and J are unit vectors In the x direction and the y direction so same same kind of representation on the plane is in space Now I will talk about dot product and cross product and Dot product also is perfectly well defined and on the plane Formals are very similar, but I will not write them They will be easy to derive by using the formulas in three in the three dimensional space a cross product the next operation that will introduce Only makes sense in in space not on the plane. So we'll only work with it in space Okay, so what's across? What's the dot product first first comes a dot product so you've got two vectors and A dot product is a certain operation Okay, it's a certain operation where you're given two vectors and you produce a number So this is very important. You start with two vectors, but you get not a vector, but you get a number This is different from say taking the sum of two vectors the sum of two vectors is again a vector You start with two vectors you get the vector also a Scalar multiple or multiplication by a number c times a You start with a number and the vector and the end result is a vector So when you think about all these operations addition scalar multiplication Dot product cross product the first thing you should remember is what is the input and what is the output usually has two inputs and one output Like Some addition to inputs both are vectors output also vector right so dot product Likewise is an operation and you know you can think of it as a kind of a black box. It's a black box. There are two there are two inputs And there is one output So there is some rule something happens here So it eats two vectors and spits out a number. Okay, so The two inputs are vector and another vector vector one and vector two and Output as a number So I have to give you the rule how this black box box works And here the important point is that there are two ways to represent this rule Which is actually very nice because then you can compare them and you can derive some useful information about vectors Which will use so what is the rule what is the rule for the dot product? so the rule number rule number one is He is Now I would like to draw to draw them on the plane or in space in space So I well I always draw everything on the plane because that's all I got I got the blackboard and thank goodness because you know if I think I would be much more difficult but Now I have these two vectors a and b And I would like to give you a rule how to produce a number out of them And so there are two rules the rule number one will only use geometric information about them So it will be kind of geometric formula so What I'll use is the magnitude of the two vectors You see this one has a magnitude which again is just a length of this pointed interval And this one also has a line has a length of magnitude sometimes I call the length sometimes magnitude It's the same thing And there is one more piece of geometric information namely the angle between them Now it's a sharp angle, but actually in general it could be up to the angle They could even be opposite to each other, but anyway, there is an angle between them So the dot product first of all the notation for it is if this one If this one is a and this one is B. This is called a dot B Which is why it's called a dot product. All right What's what so what is it? It is the length of the first one Times the length of the second one times the cosine of the angle between them This is a vector. This is a vector. This is a number. This is number This is a number. I take the product of these three numbers. So this is a number as promised I start with two vectors. I get the number. That's the dot product rule number two up to now We have not used Algebraic representation of a vector an algebraic representation of a vector Has two, you know, we have two different ways to to package this information about the vector by using these three numbers X zero Y zero and Z zero which is obtained in this way So let's suppose that a is X one Y one and Z one let me Let me write it on a different board so that I don't have to then rule number two Might as well right there rule number two is that a dot B is X one times X two plus Y one times Y two plus Z one Times it in other words. I take the X coordinates and multiply them I take the Y coordinates and multiply them and I take the Z coordinates and multiply them. It's very easy Just like this multiply multiply multiply at the mouth that's the number I get and It's sort of it looks terrible. It's a miracle the fact that the two definitions two rules are actually equivalent to each other And but it's actually very easy to prove It's very easy to prove that this implies this By using this presentation Actually, maybe it is actually might might might actually be worthwhile worthwhile doing this to give you Because actually the funny thing is in the book they explain it in the opposite direction From rule two to rule one, which is much more difficult So I think it's actually much better to To take the rule the first rule as a definition and that's often the case that the best definition is geometric because it has a nice interpretation and Usually has a much deeper meaning like here It's something which has to do with the structure of the vectors and the position of the vectors The way they are situated on the plane or in space Whereas the second definition is algebraic and it's really a working definition It's something which is very useful in calculation What might not be so useful for conceptual understanding of what we are calculating So usually the right way to go is from the geometric definition to the algebraic definition And that's the way it works here So you see and I would like I think it's a good idea to actually do this to explain this to you because here I will illustrate different rules of of Dot product in regards to addition of and multiple scalar multiplication of vectors So you see here's what I'm going to do. I'm going to rewrite this in the second way. I'm going to write it as x1 I plus x2 y1 j plus Z1 K and Likewise for the second one so far. I've done nothing. I'm just using an equivalent notation But but now it's my it's it will see how that this is actually better because what I'm going to do is I'm going to calculate the dot product By using the second definition Okay, so now it looks like a product of two things and it's tempting to open the brackets The way we would normally open the brackets if we were calculating with numbers Now a priori is not clear that this is This is allowed But in fact you can you can justify this by using by using rule one so the point is that actually We are allowed to open the brackets in the same way as we open the brackets when we do calculations with numbers So when we open the brackets, we are going to end up with Each of these terms on each of the terms in the first some dot one of the terms on the other side, right? So this is going to be x1 I dot X2 I Now in principle, I have to write down how many three times three nine terms Okay, but I'm going to save I'm going to save some time. So I I'm going to to see the following that There will be terms of two different kinds in one of them. They will have matching vectors like I dot I There will be three of them X1 I times X dot X2 I Y1 J Dot Y2 J and Z1 K Dot Z2 K Right, so these are the three Summons that I will get which will involve the same two vectors I I JJ K K and Then I will have Cross terms I will have terms which will involve say I and J or I and K and so on So, let me just write the first one of them X1 I and you will see why I will not need to write the rest of them X1 I say times Y2 J And so on so there will be I mean I'm skipping five more terms, but all of them are going to be cross terms So now I have to calculate each of this. I mean to calculate each of this Terms each of these nine terms separately and for this I will going to use the following rule that if I have X1 I say dot X2 I I can pull out the numbers I can pull out the scalars out I Can pull out the scalars and put them outside of the dot product Right so that's the same as X1 X2 Times the dot product of I and I So what's the dot product of I and I I dot I? Here is my vector I And I want to take dot product with itself Now the rule is that this is going to be equal to the lengths of I times the lengths of I Times the cosine of the angle between them, but the angle between them now is zero Right because I take the same vector twice So this is one this is one and the cosine of zero is one so this is one the dot product of I with I is one So the net result is X1 X2 So this first term gives me X1 X2 likewise the second term Y1 J Dot Y2 J Will give me Y1 Y2 Because I will need to calculate the dot product J dot J, and that's again one because J is a unit vector same calculation and finally Z1 K Dot Z2 K Will be Z1 Z2 So the first three terms will give me exactly the answer of The which appears in rule 2 X1 X2 plus Y1 Y2 plus Z1 Z2 What about the cross terms? Let me calculate the first one of them X1 I dot Y2 J That's X1 Y2 times the dot product of I and J And what's the dot product of I and J? That's zero right and the reason is that now the angle between them is pi over 2 and The cosine of pi over 2 is 0. So when I apply rule 1 I see that I dot J is 0 So all the cross terms Will disappear and I will end up with rule number 2 So you see it's very easy to go from from rule 1 to rule 2 this way And now the main point is that you can actually use you can put these two rules together and This will allow you to find the cosine of the angle between two vectors when you just when you just know once you know the The three components right because you can turn this you can turn this around and You can say that the cosine of the angle can be written as a ratio of the dot product What is this time I Think I have one more minute Okay, well you got the idea we can turn this around and calculate the cosine from knowing the length and dot product Okay, we'll continue on Thursday