 So one thing that's not on there, which I announced whenever it was I announced it was Monday. So I announced Monday was that in general, homeworks will be due in class on Wednesdays and you'll get them back on Mondays in recitation. You don't have recitation next Monday because it's Labor Day. So this time you're on your own for the homework. My office hours I think are on there. There are Tuesdays from 1.30 to 3.30. And then also Friday mornings from 11 to 12 over in the math on the P level of the math building where my primary function is to advise students who want to be majors or mess with their classes or whatever. But I'm there so you can talk to me if you have a question. We don't do that anymore. So the calendar has to be readjusted so that there is exactly one Monday holiday, one Tuesday holiday, one Thursday, one Wednesday, one Thursday, one Friday so we don't have to shuffle days. So next Tuesday is Labor Tuesday because we need a Tuesday holiday so it's the one after the Monday holiday. So there won't be any. Well today your calendar says it's Tuesday but it's really Friday. None of that is here. Other questions? Nope? Okay. So as I said before I'm trying, I'm going to take this. I will hope that this will function properly. I'll put it up on the website. It will probably take at least a day and probably a couple of days for me to edit it and do the jumps I need to do to make it web-able. So if you want to hope that it will work, that means that you don't need to take as careful notes as you would otherwise or you could do what I do and that should never take any notes and then read the book and listen and think. I strongly encourage you to pay more attention to what's going on and work on understanding as it's going rather than trying to get down everything I say because most of what I say is in the book in a different form and I hope there will also be the video of it. So it's much more important that you focus on following what's going on rather than taking careful notes. For some people taking careful notes is how you follow what's going on so that's fine. Whatever. I'm not shy. Don't come over to my house and take a picture while I'm in the shower. Any other questions? There is a homework assignment. It's not really long. I don't think it's really hard. That is due one week from today and yes, that's like seven days, whatever, seven times 24, that many hours. You could do the arithmetic better than me. Okay. So I'm going to pick up, remember what I'm doing? I'm going to pick up roughly where I'm going to review not at all so if you're screwed to that, haha. So if you remember what we're trying to do at this point is we're trying to recast some of the standard notions that you were familiar with in algebra slash Cartesian geometry, whatever so that it works in more dimensions. And what we're doing here to generalize, say the notion of a line which in two dimensions we can write as y equals mx plus b if you want. I don't actually like that formula so let me not do it that way. Let's write it as y minus y naught equals m times x minus x naught because that has a little more meaning. So this is a review. So anyway, so that means that we have some point x naught y naught that sits on the line and we can recast this in a slightly more general form where we think of this as really saying, this is really saying go to the point x naught y naught and move off in the direction that goes up m units where every one unit can go over and that's just a direction and just goes some distance in that way. So this is just saying the same thing in a different way. We go to this point and then we move in that direction some amount and that gives us everybody on the line. So this red thing can be viewed as the vector which I could write this way, let's see that's the y I could write it as a column vector like this I won't do that mostly because it's tedious which I might also write it this way which says that when I go one unit in the x direction I go m units in the y direction so either way and really let's just think of this as some vector v and so now I can rewrite this line as saying start at the origin go to the point x naught y naught and then move some distance in the direction of the vector v so every point on this line is really can be viewed as points of the form x naught y naught plus some number t which is how far I go in the v direction this is the same thing just another way to view how this line can be described so if I want to think about this point over here it's go there and then move it looks like about four times the length of that vector here and if I want to think about this point over here on that line it's go there and back up about two and a half times the length of the vector so that describes all of these points these are of course vectors but we think of them as vectors that point from the origin to that point on the line and the advantage of doing that instead of using this formula is that that notion works just as well in any number of dimensions so that if I'm working in three dimensional space and I've got some point here sitting out in three dimensional space given three coordinates x naught y naught that's supposed to be z z naught I can play the same game I say go to that point as this vector and then from that point pick some direction that you want to go about this way and then just take everything that moves in the same direction as this vector by some scale of that so this line which is a little harder to visualize in three dimensions I guess it punctures the plane just about here so that's below in my visualization of what's going on here this describes that line because it just says go here and then move that way by some amount of t so if we call this vector v again then these guys are exactly the same form this line is the stuff of the form in fact let's call this guy I don't know a this is everything of the form a plus t times v and this will work in any number of dimensions if I'm in a 17 dimensional space which is a little hard to draw I will have a vector in r 17 plus t it's a direction in r 17 and that will describe everything on that line let's go there and then move off that way and this is sort of naturally how you might think of a line as well go down to the beach and aim your camera that way and what do you see? everything there it's exactly the same okay so this is what we did last time let's just see if anybody understands so what's the equation of the line through the point let's take two points 1, 2, 3 and 4, 5, 6 so if you understand you should be able to answer that question or at least walk me through how we do that so don't just tell me the answer tell me what you do yeah well first you want to find that that is between the two points okay so let me draw a cartoon of what's going on here's 1, 2, 3 here's 4, 5, 6 and I want to find the vector between them okay so what's the vector between them? get that one okay so that should be like 3, 3, 3 wow okay that's just amazing how that worked out so this vector is the vector coordinates 3, 3, 3 now you can use either the two points is your initial point okay plus t times the vector 3, 3, 3 okay so this will be let's use 1, 2, 3 1, 2, 3 plus t times 3, 3, 3 good anybody confused about that? now we can write this in a slightly different way we can put them together this is 1 plus 3t 2 plus 3t 3 plus 3t so if we don't like vectors you can see that we are actually describing this line by telling you all x coordinates of the form 1 plus 3t y of the form 2 plus 3t z of the form 3 plus 3t so we can also describe this instead of in vector form as x is 1 plus 3t y is 1 2 plus 3t and z is 3 plus 3t so those three equations together also are another way to describe this line it's the collection of points that look like that these are really exactly the same just written out in a slightly different form so you pick the t value and then we know where we are in the line there's actually a little more information here than just the line there's sort of an implied speed that if I increase t by 1 unit then these guys increase by 3 units and so this is the same line in terms of the set of points this line let me write it in vector form 1 plus s let's use t they're not the same t 3 plus t 3 plus t they're not the same t so I really should use I don't know, t1 I don't know why I'm using t but I am this is the same line in terms of points but the speed is 1 third slower if I increase t by 1 unit I only move a third of the way to 4, 5, 6 whereas here if I increase t by 1 unit I move all of the way to 4, 5, 6 but since we're only paying attention at this point to the collection of points that are described by this these are the same line just like you can drive down the LIE at 30 miles an hour you'll probably get her ended, but you can or you can drive down the LIE at 80 miles an hour you're traveling the same path but you'll get there a lot quicker at 80 miles an hour unless you get arrested so you can do either one and they're the same path but at different speeds and so sometimes you care about the parameterization the way you're describing this collection of points and sometimes you just care about the collection of points so here the speed is different and of course I don't need the same base point either I can use 4, 5, 6 as what's your name? Nick as Nick said I will maybe remember next time I see you that your name's Nick but at least forgive me if I don't but maybe by the end of the semester I'll remember unless you leave okay so any questions about lines stuff with lines no okay so in two dimensions that's about all we can deal with but in three dimensions we can also think about planes so let's think about how to describe a plane so these are I'm only dealing right now with flat objects I'm dealing with the most simple geometric objects like lines and planes and hyperplanes and so on but so let's think about how you might describe a plane how many are is anyone not familiar with this notation you're not familiar okay so I will use this notation because well I used to it so the line so this is on the side the real line we use r for the real numbers because r go together and r2 is a copy of the real numbers in each direction and we consider all pairs of those so r2 is the plane which is pairs of reals and r3 is obviously space r12 is 12 numbers the set of all x1 x2 up to x12 such that x is important so it's all 12 couples of numbers so this is useful notation if we want to specialize to a specific dimension we also don't have to think about just reals in this way we can think about integers in this way so I don't know that I'm going to use it but let's just say it anyway z is the integers which is all the negatives 0, all the positive numbers someone and say z2 will be pairs of them and we can generalize this in many ways so also we'll do a little complex notation z is the complexes so this notation is fairly standard in mathematics but you cannot see it until somewhere around the university or college level high school you pretty much have to deal with real numbers or you deal with integers what the difference is good, it's a fraction thing let's do that q is the rational set of fractions q isn't 0 what else I guess they have no common divisor about the gcd z is for zealot and german and q is what is q at the bottom I don't know it doesn't stand for comptabular so those are the common ones that we'll use and typically we'll focus on rn n is not a half n is a positive full number although r0 looks like it's 0 real numbers not much to do with it but we probably won't deal much with r0 that's the wrong set let's go back to this what about a plane in three dimensions how can we describe let's draw a picture I like to draw pictures you can't read my pictures but I can so here's a plane it sticks in the back it doesn't look much like a plane anyway I'll try it I know what I'm drawing I don't care if you don't know actually so here's a plane and it's in r3 so there's a plane sitting in r3 somehow generalizing this notion how might we describe a typical point in the plane here right so I need a point in the plane that isn't this one I need to know how to get to the plane in the first place and then I can get to this point by just taking this vector but that will give me a line sitting in that plane so I need another vector to get everything else and then I can express that as a linear combination of these two vectors just like if we want to describe point on the map we have two coordinates tell us how far north or south and how far east or west here we can say how much in the green direction and how much in the red direction so I can get to this point here by saying go to the black point move in the green direction a certain amount and then move off in the red direction it's the same as that red direction it's parallel so direction doesn't have a base point it just has a way to go and these are supposed to look like the same one and maybe this is half of a red but whatever so one thing though we have to keep in mind is I could be stupid and choose my red direction to be parallel to my green direction and then I'm not going to get much of a plane so to describe the points in my plane I need some kind of a base point so I have my base point let's call it P I don't know why I switched from A to P but I did and then I have two dimensions to add on I have one direction that I go in the T and then I have some amount that's my, in this picture is my green direction and in this other way I have get somewhere on the plane move in the green direction some amount move in the red direction some amount and I need V and S no W to be not parallel if V is parallel to W I'm not going to get a plane I'm going to get a line because then I can collapse these two together and describe in terms of T plus S divided by the dimension but I can just describe this as a line this describes always a two-dimensional plane in any space that is three or more dimensions I mean it kind of describes a plane in the plane but there's only one so there's not a lot to do we can describe something in R2 by this 2 but every point in R2 is a point in R2 so we always get the same plane I have no idea if you understood what I meant by that but what I'm saying is if we're talking about it in R2 we can describe points in R2 this way as well I see it as a second thing take a base point go some red direction some amount and some green direction some amount and for as long as the red and the green guys live in the same plane any point on the board this way but every point in the board is always represented this way there's only one board to put it in okay I answer that did I answer it? no share do you have enough gum for everybody? I was just thinking about whether or not you could have a two-dimensional plane in four space yes you can so this will work just fine two vectors with four components and this will describe a two-dimensional plane in four space but the complement of that is also a big space right just doesn't divide four space in half in the same way that a line doesn't divide three space so if you wanted to do space three space exactly you would have three non-parallel vectors and if you wanted to describe some eight-dimensional space we have eight non-parallel vectors so depending on how much you have you can still have three non-parallel vectors yeah so when I meant non-parallel I meant non-planar non-planar non-parallel is wrong non-planar so I need them to be so the correct word which since we're doing the linear algebra here is that another way to say this is that V and W are linearly linearly independent and then the generalization works so if I have three vectors I need U, V, and W to be all linearly independent I didn't define what linearly independent means but you can't do the others yeah I can't express one in terms of the others so another way to say this so let's go off we'll get there by the end of the year so this is not yet what we're really doing but what the heck I'll tell you now so some collection of vectors V1, V2 up to the end so I have n vectors and these guys are linearly independent if we have let me write it this way A1, V1 plus A2, V2 plus up to An, Vn equals zero there's an equation if this is true means that all of the a's are all zero so the only way that such an equation can be solved is if all of the coefficients are zero then these guys are linearly independent so this is usually the standard definition for linearly independent but a something that you need to think about to see that they're the same and when we discuss this I'll go through it again this is the same thing as I can't write An, V in terms of the a's so this is a more mathy way to say it if you think about this for a few minutes or a few seconds or a few hours depending on how long it takes you to puzzle things out you'll see that these are the same and I can explain it to you in its few seconds but let me not let me leave it for you to think about this and you decide whether it's going to take you a few seconds or a few minutes or a few hours wait a few weeks until we cover the puzzle okay so that was my side remark so in general can't read what I wrote that means that is it guessed yeah do you have an arrow over the zero this is zero vector these are vectors you have arrows over them and this is a vector I'm going to miss these little vectors a lot so I should write them in boldface I'll take all the next A, V and V I don't think so okay so this is the definition of linearly independent don't worry about it if you didn't follow it it really means this so coming back to plane so if we have a hyperplane I don't know a K dimensional hyperplane is going to be and obviously it only works in a bigger than K dimension otherwise it's stupid a K dimensional hyperplane is going to be something of the form some vector P plus T1, V1 T2, V2 blah blah blah K, VK so I forget who asked about 4 plane or something so here we can describe a 4 dimensional plane in this way and here K is 4 so we stop at the end yeah up to K minus 1 no I have K because I started at 1 instead of 0 if you want me to start at 0 I'll go to K minus 1 my math is not computer science so I start counting at 1 he asked if it should end at K minus 1 but I need K vectors so if I start counting with 1 I stop at K to get K things if I'm a computer person I always start counting at 0 and then I stop at K minus 1 and if I'm really weird I might start counting at 7 and go to 7 plus K I mean that's how many numbers so when you say plane it's not necessarily a flat plane it could be like our space no it's a flat plane nobody call it a hyper plane because it's more than 2 dimensional so generally in English one calls something a plane it means a 2 dimensional space and so a hyper plane that's a K dimensional space because it's in a hyper space yeah it's in a hyper space the plane itself is more than 2 dimensions so it's not a plane in the sense of this floor I can think of this room as a 3 dimensional plane sitting in 4 dimensional space or a piece of a 3 dimensional plane where I'm in 4 dimensional space where the 4th dimension is time and if I just take a snapshot of this room at 1 instant in time that's really a hyper plane a 3 dimensional flat space sitting in 4 dimensional space so once it's it is a 3 dimensional plane you call it a hyper plane or a 3 plane yes hyper plane usually means a plane other than a 2 dimensional plane in 3 dimensional space so if you don't use the word hyper that's fine plane always implies flat so if it's curved it's not a plane is it curved? are we good? okay yeah this is in so this is in r m and here m is greater than or equal to k and each of the vectors v1 up to vk are all m vectors let me write it in column notation 1, 2, 3, 4 plus t times 7, 1, 6, 5 plus s times 8, 0, 7, 9 plus uh I need another letter r times 1, 1, 1, 1 this is a 3 plane in 4 space and if I add another number here 5, 5, 5 5 this is a 3 plane in 5 space but this 3 plane does not necessarily is not necessarily to be a flat plane what's flat mean? I understand that this is flat in an easy to make sense of way but in the same way this room is flat one can define and we're getting way ahead of ourselves the notion of curvature in more dimensions than 2 more than a curved thing we can define curvature we can talk about a curved space in a well defined way this is not curved at least not in a metric that we've written things it's not flat in the sense that you can ride your roller skates on it because it's a 3 dimensional thing it's a 3 dimensional thing sitting in a 5 dimensional world right so you can ride your roller skates and fly your airplane that's fine but you can't use your time machine yeah so if you're to square any one of those terms like t squared we'll have a different discussion it may or may not become curved hands on and stuff so we'll talk about that more but for now we're just thinking about the easy stuff the flat stuff just like before we started doing calculus here you think a lot about lines for I don't know from junior high school to early high school you think about lines and only after a while you say hey it could be prevalent can you talk about circles okay so we have that and I'm like when do we stop 5.20 so I'm not waiting I'm just a bill alright so we have these geometric objects and here lines and planes are kind of the same kind of thing just a matter of extra dimensions and things like that so for the next class or two we're going to mostly focus on lines and planes and then we'll move to more complicated objects once we've mastered that so but actually let's step back a little bit let's talk some more about vectors so if you remember I can add vectors and when I do this I just add the subtractive components or geometrically if I have a vector v1 and a vector w if I want to add them together I just take the v I stick it on the n's w and then this new vector v plus w which just corresponds to adding together the coordinates which is just saying go in the w direction until you get to the n now travel in the direction of v you get to its n and in the n you've gone to v plus w so this is fine I didn't mention but of course somebody already mentioned for me we have a special vector called the zero vector which is just all zeros and so we have some algebra of vectors we can add them subtract them vectors have inverses we have a zero vector yeah because it looks like this so the zero vector is don't going on and here I'm being a little sloppy I'm implying that we're at a fixed number of dimensions there's only one zero vector for each dimension just like there's only one vector one one one can you also say that any vector has an ending for a zero from the end of it so it's the same thing you can invent the vectors in a higher dimension by sticking zeros on the end but what I mean by the zero vector so let's be a little more algebraic about this so we could I could write all of this stuff in the way that if you were taking math 310 or maybe even math 2.11 where we could write some properties which describe vectors in the way they operate together one on the other we're saying that we have some vector V so I can give an axiom that says that for any vector V there there's another vector let's call it negative V so that V plus negative V and there's a unique vector zero so I'm just saying that I have lots of vectors one of them is special I call him zero and anytime I have a vector he has a friend who I call negative V so that when I add them together I go to zero this is similar to say in the plane there's an origin there's a special point in the plane which is where the axons cross and if we think of this as it's a collection of all vectors that's the zero vector now it doesn't necessarily have a unique representation in that I can write it as 111 minus 111 still a zero vector but I have a special die called zero so it would be proper to say that 11 plus negative 181 big zero zero with an arrow or bold zero so often I will be sloppy and everybody will be sloppy and Nick may be cockney on this I just wrote zero and I didn't put an arrow this is false this doesn't even mean anything it needs to be a vector for it to mean something because if I add up a bunch of vectors I better get a vector back I can't add two vectors and get a dog I add two vectors and get another vector so there's a lot more properties that describe vectors like given two vectors if I add them together this is another vector so just like numbers if I take two numbers and add them together I get another number this is true of real numbers but it's not well it's not true of numbers less than 10 if I add together two numbers less than 10 I don't get another number less than 10 necessarily I might sometimes but it's not closed under addition but vectors addition of vectors is closed and I can describe it in terms of the little arrows I can describe it in terms of writing in coordinates and add the coordinates together they're the same thing so if I add two vectors and it's also it's the same as W plus V that is vector addition is community now I don't need you to memorize all these properties but I'm just writing them down vectors work the way you would expect but then in addition to having that as I mentioned before we also have another collection of numbers laying around so if you take 308 we'll definitely get back to this in a more formal way I mean kind of informal about how vectors work now but we have in addition to our collection of vectors we have scalars so the scalars they need to be in a field and in our case our field is real and that is given a vector V some vector V in my vector space and given some scalar space and given some scalar space and given some scalar space and given some scalar space and given some scalar R in the reals then our field is a well-defined vector G and it works the right way we can prove these facts but I'm not going to and similarly there's a special scalar so the number one times V is joint V so we need that if you multiply by one nothing changes otherwise that's kind of how one should work and zero times V should be the zero vector now these properties are not all will be independent but don't worry about it what else do I need and I have a distributive law right if I take R and I take S as scalars and I scale V by R plus S this is the same as just scaling V by R and then scaling V by S and adding the answer together that works too and similarly if I scale with some vectors that's the same as scaling independently I really hadn't planned to write down all of these properties that give us a vector space I sort of got lost anyway so we have that so we have these vectors I'll finish them by factors I guess I neglected to put subtraction if I want to subtract one from the other then I go it's the other diagonal that's the difference now it's a little hard to see let's see why is it now so if I put negative this is V minus W because I put so this is V minus and the other diagonal is the difference what you can see by just taking W flipping it around sticking it there if I take this W with minus W here with that on the there you're lost? Me too yes it doesn't make any sense so this is not defined for us we can't multiply vectors we will have two operations and I'm getting to one right now that work like multiplication but they are not multiplication in the same way that 2 times 3 equals 6 is multiplication they just share many properties of multiplication and we use the same symbols but they do not mean multiplication if you think about it I have an arrow and I have another arrow and I want to multiply that what does that mean? it doesn't mean anything I don't know how to multiply arrows I know how to add them because I just picked this one up and put it on the end that's added but I don't have a good way to make good sense of multiplying I mean there are operations that look kind of like multiplication and we'll talk about them very soon that give us either another arrow or give us another but yeah this doesn't make sense so in terms of arrows everything I wrote here makes sense adding them means stick one on the other scaling it means stretch it by that factor so if we just think of it as little arrows this makes sense and that doesn't but now that I said we can't do it I'm going to do it but it's not the same as what two times three means so one thing you might try to do suppose I have a vector v which is 1, 2, 3 and I have a vector w which is 1 minus 4 5 you might say well let's multiply this stuff together so let's just take this times this, this times this this times this well then let's add them together just for fun so this actually means something it's kind of a way to multiply vectors but it's a number right this is 1 minus 8 plus 15 0 8 0 8 it's one of those numbers ok so this is a number and it's a number that has something to do with me and w I can define this it makes sense and if you just write this down and I'm just going to call this v dot w because it's kind of like multiplying and maybe I should use a little x so this is something called the dot product multiply the coordinates together and add them up but it means something in terms of the vectors and I want to, so let's look at a few properties of this guy before I try and interpret what it means well if I have a vector v so certainly it's defined for any two vectors I write down so it works for every two vectors and if I have a v and I dot product it with a w it gives me a number it will give me exactly the same number as if I do live in the other order the order doesn't matter because multiplication is commutative and different and addition is commutative so if I change the order around it doesn't matter I get the same answer so this is symmetric what's another property that this has well let me tell you because I have it in mind if I take any vector and I multiply it by itself in this way what can I say about it this is like v squared but it's not v squared it's always positive except for the zero vector so if v is the non-zero vector then the dot product with itself is positive and the zero vector the dot product with itself is zero so the only vector where the answer is zero by the dot product with itself is the zero vector now that's a useful thing actually because well let me finish this we also have kind of a distributive business if I take if I take so notice that this doesn't make sense v dot w dot z this doesn't make sense because this is a number and the dot product is only defined on pairs of vectors so this is meaningless yeah I think vector v times vector of w is a number times another vector should be another vector that's right these are dots so dot means take two vectors give me a number I can't take a number and dot it with a number so this dot so this means something if this isn't this this becomes a scale which then I can scale it but this is not the same as this so this kind of doesn't make sense once I combine two vectors I don't have another vector to combine them with I don't have a scale so I can make sense of v dot w times z that makes sense that means stretch z by the dot product of v and w but I can't make sense of v dot w dot z so it's important to remember what are the domains of these operators these dot things these vectors and it gives you a number ok but I guess coming back to this for a minute the way I defined it this has an inherent geometric meaning I have some vector x and I dot it with x this gives me a number let's call it a gives me a number what is the relationship between number and this vector a I mean vector x it's the length of x right this is the length of x let's see why that is so I have some vector x which has coordinates x1 x2 let's just stop it and if I think about let's do it in 2 it works in any number but let's just do it in 2 x1, x2 now that means that this direction here is x1 and this length here is x2 and by the Pythagorean theorem this length is the square root of x1 x2 x1 I don't know what I'm doing x1 squared plus x2 Pythagorean theorem which is well if I take x1 x2 and I dot it with x1, x2 I get x1 squared plus x2 squared so this is x dot into the x squared so we can pull out the length of the vector by this dot product thing and this statement is just a way of saying every vector has a length except the zero vector has a positive length and the zero vector has zero length so this statement says just what I said every vector has a non-zero length except the zero vector this is really the square of the length but ok again let's and it should be apparent to you that if we're in free space you can use the distance formula and we'll get the same answer if we're in 12 space use the distance formula for 12 dimensional space but this is sort of a convenient way to encapsulate the distance formula to length of the vector and another property suppose I have two vectors playing around I have y here I have y and here I have some vector x and so let's do the simplest case or the perpendicular let's start with that case first so this is the right angle the way I drew it but let's start with this case and now what will x dot y be? well so if you know what the dot product is you know it's zero but let's see that it's going to be zero so how can we see that? how many people already know about the dot product? so like I'm wasting all this time explaining how you think well those of you that don't you listen you guys go to sleep okay so let's calculate x dot y in this case and actually I don't want to do that okay so I plan that this is zero and well in this case it's sort of particularly easy because I drew it in two dimensions I can write this as I put my x on the y axis and my y on the x axis so this is the vector y zero and this is the vector of zero x and so when I calculate their dot product x they have little hats x dot y is going to be zero times y plus it's not a vector zero times y okay so now in higher dimensions it's still going to work the same way but let's bend it a little bit so if I have some angle between them well so this vector here is the vector x minus y and I want to somehow relate x minus y to x and y and theta pretend for a minute that we're in a geometry class we don't have vectors that's not a right angle it looks like one this is some angle theta this is some a this is some b this is some c does anybody remember something about the relationship between a, b and c? the love of time so what does it say? be sure it was c squared equals b squared plus j squared minus which would really be cosine and z and that would be right so this is the law of cosines and it relates to the lengths of the sides of a non-right triangle or a right triangle because in a right triangle theta is 90 degrees so the cosine of theta is zero so this term is gone and we just get the Pythagorean theorem as a special case so here we have this is my c and then I get x squared plus y squared c squared and it's the same because 2xy cosine of theta, theta is 90 degrees so that term is gone so we have this and I want to somehow relate this to the dot product to pull out something about theta so in this picture where a is x a is y and c is x minus y let's just rewrite the law of cosines using those vectors well they're not vectors really I'm just thinking about the lengths I'm sorry c should be y probably no c is x minus y in this picture yeah did I mess up my okay so I just want to plug this in where this is c this is a this is b so it should be true just from the picture it's a triangle so it's a triangle so it's still true that x minus y square should be equal to x squared I've permuted the order I'm sorry but things are commuted in the right way so I'm lucky minus 2ab sorry 2xy cosine theta so that should tell me something and I want to try and pull out something about theta in terms of the length of x and the length of y and maybe the dot product or something like that okay so now I'm going to write x minus y squared in a different way I'm sorry I'm going to write it this is a vector it's the length of this vector so the square of its length is just the dot product of the vector x minus y that's a vector and then when I started writing down the rules and didn't finish them I'm going to use them so I should have written them down but you bet so we have a kind of a distributed law here this is the same thing I can say well that's going to be x dotted with x minus y that's okay minus y dotted x minus y those are pairs of vectors so does everybody believe that that works yes you know just writing symbols so whatever okay and then I can play it again x dotted with x two vectors minus x dotted with y minus y dotted with x minus well it's plus because I have minus minus y dotted with y and then I can rewrite this in terms of length this is the length of x square this is I'm going to reorder plus the length of y square and then this is minus two x dotted with y okay well now comparing this to the law of cosines that tells me this is a vector which is the opposite side of this triangle this is the length of the opposite side of the triangle this is the length of one side this is the length of the other side and this is something so looking comparing this to the law of cosines I see that it must be true there's no other way that this and this are equal but they are because those are the same thing is if x y cosine theta is exactly x dot y there's no other way that this will be true so I've just calculated the cosine the angle between them is the dot product divided by the product of the length this is a very important calculation of the pool and so the dot product which more formally is called the inner product but the inner product of two vectors tells us the cosine of the angle between them doesn't quite tell us the angle close but in particular you get that when is the cosine zero the cosine is zero exactly when these two are orthogonal so we also get that x dot y is zero exactly when x is perpendicular to y so this dot product thing is a very useful object it allows us to measure angles and lengths and in particular tell when two things are orthogonal and notice that all of this calculation didn't use anything about coordinates it just used properties of vectors so this this calculation although I sort of drew the picture in the plane even if we are in three space or seven space there is a plane that goes through two vectors so there is a well defined triangle so all of this stuff works in any dimension it didn't do anything bad it doesn't generalize to any number of dimensions so we have a well defined angle between two vectors and we know how to calculate it we can tell when two vectors are orthogonal no matter what no matter how many dimensions we are in and that's a really useful thing okay so I have a way to do lengths so if I have a way to calculate lengths this means I have a way to define length one so the dot product measures angles and gives lengths and in fact so I'll just say these words so we can generalize instead of using a dot product more generally we have something called an inner product which in the Euclidean in the standard Euclidean space that we are working with in this case standard Euclidean space that we are working with in this class this inner product is the dot product but we can make sense of vectors that aren't arrows in three dimensional space vectors are a more general concept of polynomials of vector so if you have an inner product you have a way to measure angles and lengths so there's this notion you might have heard the word Riemannian geometry or Riemannian metric anybody heard this word before okay sorry R I E M A N N you've seen his name before yes his name wasn't E yet his name was Bernard so Riemannian geometry this means we have an inner product so we can generalize this notion of flat spaces to curved spaces and the inner product or the way we measure angles between things tells us how the geometry works how the space works the geometry that works in the normal sense this is not a course on Riemannian geometry although Riemannian geometry is a generalization of a lot of what we're doing but if we generalize this notion of inner product we get a more general kind of geometry than just flat geometry and so there's a lot of people at Stony Brook that study Riemannian geometry not Riemannian geometry okay so that was just I don't know me saying stuff okay so a unit vector sometimes we might want to just think about directions and not worry about lengths so if you think about like a compass point this is supposed to be a compass so we have a compass here and we just have an arrow that moves around and gives us a direction or in space or whatever and we don't care how long the arrow is so we might as well always make it 1 so we can think of a unit vector it always has a direction and it's very useful unit vectors have this property that they don't when you use them they don't sort of stretch things so just to make a unit vector set the zero vector and we just divide it by its length this gives us the unit vector u which is just sort of a pure direction because the length is 1 so they don't care about the length so a lot of the formulas become a little simpler if we're dealing with unit vectors but I want to mention the idea of unit vector is a lot of times the word of unit vector comes up let's just take the vector divided by its length it's 1 so the length of u is 1 3 minutes so another thing that we often want to do with vectors is you might want to cast the shadow of one vector on another you might want to take the component of one vector in a certain direction so I might have a vector here v if I look in this direction how long is the shadow of v here so this is maybe let's call it w because I don't necessarily want to specialize to a unit vector w so I want to know how long is the shadow or the projection in the w direction so this is a question that is very useful maybe it's obvious at this point why it's useful but it is useful for doing things like calculating distances between two lines or distances between planes or stuff related to calculated distance projecting one thing onto another is a useful thing so I claim so when we talk about the distance between a point and a line we always mean the minimal distance so that's the length of the perpendicular so this is the distance between the point and I want this projection which is the other component so how would we do this yeah I think it's like v dot w over w dot w times w is this because you just thinking about it or is this because it's a formula that you saw before you need to if you do v dot w you get absolute value of v times the absolute value of w and then we know in the end no matter what we're going to be multiplying by the vector w right and then no matter what we know that we're going to be multiplying by the vector w in the end in the roman direction so you'd write that up too like vector w and then we need to divide by w okay so so we have this angle here and certainly v dot w is the length of v, length of w cosine of the angle between them and but we also want a vector so and I want to think of w as a unit vector and now I just look into w direction so that's what you said this is not v dot w anymore sorry so this is the projection of this thing onto that and since I'm out of time and they're going to come in here let's stop here, I'll pick up with this on Wednesday I think there's some of this on the homework so there is a homework it's on the website