 Okay, so before I get started on the next bit, I apologize I didn't get as far as I had hoped to last time, and then of course, one thing that you maybe should have noticed is that I focused more on the ideas and less on the specific problems. This will happen. We don't have time to do 23 examples of each possible problem, so I want to convey to you in the course the ideas, and I'm happy to do them in specific problems, but hopefully the idea that the plan is will understand the concepts and then figure out how to apply them in the context of the problems. Another comment I want to make, you should have received the solar message, and I guess a few people are late because they misunderstood the solar message, which is that not today, but next Monday, the recitation has moved from heavy engineering to the second floor of this building, so that means you don't have to trunch across the campus to get to your recitation. It's, I think, 217, is that the number? Yeah, that's what I saw. So it's now in I217 and not over any heavy engineering. I'm sorry? Who's asking if that's the second one? That's correct. It's moved there, that's where it will be. Precisely because I'm kind of stupid for it to be on the other side of the campus, so I could have asked them to move to the hospital, but... Okay, so there's that. So I will give you a choice since we're just good. So one thing at the end of the, so another thing, I put the video up. It's in, but it works. So you can see if it works or not. Use it if you don't. This one will probably be up in maybe two days. It takes a lot of computer time to crunch the video into something that's usable. Or not usable, at least plays on the computer. As it comes out of there, it's about ten times the size that it needs to be. And I don't think you want to download ten-gigas video for each class. Okay, so there's that. Last time when I was talking about... So if you have homework, you can wind up in this pile at some time in the near future. So last time, as you probably noticed, I didn't really go heavily over the idea of projection and so on. I ended the class. I wrote the wrong formula on the board because I forgot about one thing. I don't want to pick up from there. Most of you probably figured it out because it was required to do some of the homework problems. And you did the homework problems or you didn't, and consequently you read the book or you didn't. So I want to pick up from there unless everybody tells me, I got it. I don't need to hear about it. Nobody's going to say it. You got it. You came to my office. So you got it. What did you say that day? Problem? What did you say that day? Okay, so at the end of the last class, I was talking about we have a vector. Let's use the same note. No, I'm not going to use the same note. I have some vector, what do I want to call it? C. And I have some vector w. And I want to compute, in some sense, if I draw up a right angle here, I want to compute either this vector or its length. And then you also had a homework problem and this I didn't quite get to. This vector as well. So this red vector here is called the projection of the C. It has other names too. In some contexts it's called the length. It's called the length. It's sometimes called the component, vector C in the w direction and so on. So there's various names for this sort of thing, but it also is useful. So I drew it parallel to the x-axis, but it's certainly useful, say, in the context of a physics problem where C is representing some force and you wanted on how much of this force is being applied in the direction, in a certain direction. So you might have, in a physics context, you might have some hill and there's a force due to gravity and you want to know how much of that force goes this way or something like that. So that's the same kind of thing, right? We're going to project this gravitational force onto the direction of the hill and we can figure out what piece of that is being contributed in that direction. So, but another context is if we have, we want to change coordinate vector, change coordinate somehow. I have two coordinate systems. I have my original coordinate system and I have some new coordinate system here and I have some vector in the green coordinate system which I know in terms of x and y and I want to express this vector in the new, well that's black, I'll use the black coordinate system, in the black coordinate system in terms of some, let's call it u and v. So I might want to change coordinates from the x, y space to the u, v space and this is a big piece of linear algebra it has to do with coordinate changes but this is another, these are the same question really and so finding the component of this in that direction is just sort of telling me how to express this thing which didn't move the green thing in a new, in a new coordinate system. Okay, so I didn't calculate this I sort of did it at the end of the last class you did it on the homework or you didn't go to the other maybe you understood what you were doing on the homework maybe you didn't, I don't know, could I do it? Should I assume you all understood it? Alright, then we'll never get anywhere Okay, so at the end of the last class so let's, I'm going to redraw this picture so here's my w, it's the long guy here's c and the reason I use c is so that I can think of this as a triangle a and c and the angle between a and c is theta and we certainly know that the cosine of theta is the length of a is the length of a over the length of c or in other words the length of a is c cosine theta and these are vectors I'm going to probably forget so this is one way to express the length of a but we also know that we can calculate the cosine of an angle between two vectors using the dot product so we also know that the cosine of theta is the length of a the length of c times a dot c so we take the dot product of oh, no no, yes, that was it sorry, writing garbage a c cosine theta and we have this equals that so we can see here we can easily calculate a dot c and what we want is the length of a so that means that the length of a is the length of c cosine theta which is the same as a dot c except I have an extra a over there no, extra c I have an extra a, this is not what I wanted oh, w, sorry I want w here I want w here because this is something I know sorry okay, so w dot c is certainly the length of w the length of c times the cosine of the angle between them but what I want is a so what I want is c cosine theta so the length of a is certainly going to be w c cosine theta divided by w because the length of a is c cosine theta and sorry I confused myself but that happens which is easily calculated because that is w dot c we know w, we know c and then we have to divide by the length of w so that tells us the length of this red vector another way you could view this same question which is sort of the way the book describes it is I could think of making w into a unit vector and if w is a unit vector so I could make some new vector which the book calls n I think that is an unfortunate choice because n usually means normal so I'm going to call it u which is just the vector w divided by its length so this is a vector of length one which I guess in this picture I've run out of colors so I'll make it black dotted there it is so this is u which is just w stretched back to be of length one and then this calculation which doesn't depend on anything about w except its direction because we divide by the length here becomes simpler because first I've unitized w into u and so when I take the dot product of u with c it doesn't contribute any extra junk that I have to divide away so this is also the same as just u dotted in c and this is a number all of these are numbers if I want it to be a vector like this red vector here then I have to make it into a vector so I could also say that the vector a is the vector u dotted with c that's a number and then I put it in the u direction or if I don't want to think about u I could also do the calculation it's exactly the same what is that line? I don't know it's exactly the same as saying it's w dotted with c divided by the length of w that's a number times the vector w but I want to unitize it small square here so either this is all the same and then this vector which I guess I call b in the picture which is the component of c in the other direction I can just do by subtracting c minus this vector a so in that picture the vector b is just well I take c and I throw away the part from the a so I can go through an example of this if you want with actual numbers none of this depends on whether it's in the plane or in 17 dimensional space all of these calculations are completely vectors or scalars it doesn't matter how many dimensions we're doing here this is all the same because once you have two vectors you have a plane that they live in and so it doesn't matter whether this sticks out of the board or however many dimensions it's in any questions on this at all you all did examples on the homework so I'm just reminding you what you did in the homework if the way you did the homework is to flip through the chapter find the formula plug in and there you go because I know that I talked to two people today that's what they did and I went through this kind of a discussion with them I don't know what they did so there's that I mean I brought the notes from a meeting I went to rather than the notes I brought them up okay let's come back to now I mean it's related not quite the same I want to return to the idea of calculating of writing equations for lines and planes and so on so we certainly saw one way to describe is just we take some point on the we take a line here's the axes here's a point on the line I take a vector to any point on the line and then I add to that vector T times some vector some vector that I can stretch so I can describe this line as this line L is all points of the form A plus T that's certainly one way I can describe a line and we can generalize this to a plane or a hyper plane so here's a plane here's my origin I take some point on the plane I take a vector to that point and now I have to add two vectors well that's a terrible number this vector and say this vector where this is a vector again A and this is some Q and this is some B and I can describe everything in this plane by saying and then take some combination of U's and V's to move around in the plane so I can describe this plane as vectors whose end point lives at A plus some scalar times V plus another scalar times U and so on I can also describe this in another way which sort of unifies this concept a little bit and before I do that I have to get used to that I have to remember where I can stand and then my life is easier because then there's less editing of the videos I think so and we can generalize this to something of the form A plus T times V plus S times U plus R times W and somebody last time asked about this plane is in flat so you and I were using different meanings of the word flat your meaning of the word flat is like a tabletop and my meaning of the word flat was not curved so this is not a curved space so I say it's flat but it's not flat like I can write a bicycle on it so there's two different meanings of the word flat one means a two dimensional flat surface and that's usually what people use in the word in English to describe a plane and another meaning which is a more geometrical more mathematically generalized thing is not bent, not curved and when I said sure this is flat of course it's flat I meant it's not a curved space so ok so let me just shut up about that ok so it's flat and it's not flat and if you didn't understand that so I want to come back to this idea of plane and say how else could we describe a plane or how else could we describe a line in a way that is sort of the same here you can see the generalization from a place on the plane and moving around on the line you can think of a line like a one dimensional plane in the sense that it's a one dimensional object where you can move around in one direction and then this guy is a two plane probably in three space maybe it's in seven space but whatever and then this guy is a three plane well I already wrote the word three plane again and we can generalize this to a twelve plane or whatever we take a starting place on the plane and then some combination of directions in those higher dimensional spaces and that will certainly describe for us a higher dimensional object you might think let me get away from what I'm saying for a minute you might think that that would be silly why do we need to describe a five dimensional plane like object but a lot of times we do have five dimensions of freedom that scale scale linearly and we need to describe those objects so for example you might be trying to describe position and velocity of a particle well that would be in space right I have three dimensions to describe the position and another three components to describe velocity that would be a six dimensional thing I have six numbers to describe the position and velocity of a particle now they are subjects usually to certain laws of motion so that will come back down you know we have conservation momentum and we have conservation we have all sorts of conservation laws but inherently that's a six dimensional object or maybe if you're describing some inputs from a pair of joysticks that's four dimensions there I have two forward no directions I can move this forward and this back so if I want to describe I have a pair of joysticks that are going to move something around that's four dimensions and that position would depend linearly on those four inputs and so on so I have many contexts in which you encounter multi-dimensional spaces that you don't usually draw you know as geometrical objects like this and there may be some restrictions on it maybe you have some conservation laws that constrain that motion but still there are many places in which you will encounter higher dimensional than three things and it doesn't mean they're not real it just doesn't mean they describe things that you can build a sculpture out of okay so back to this I want to generalize this notion a little bit and let's just start with a line in a plane and let's try and think of so here's a line and I can describe this line well I can describe the direction of this line either by giving it back through this way but I can also describe it in terms of the complementary direction okay that doesn't seem like I bought anything I traded one vector for another but if you think about a plane that's not the wrong this is a plane it's flying over the space here the things on the plane the things on the plane I can describe it as a combination of these two vectors living in the plane but I can also say it's all the stuff perpendicular to that and now you see there's a lot of consistency here this line so the line is the stuff so I need a point here too so it's the stuff which is parallel to the black vector which is the same as the stuff perpendicular to the green vector and this is called a normal vector all the points in the plane that are parallel to the black vector and pass through a given point or perpendicular to the green vector and pass through a given point and here the plane we can do in the same way it's all the stuff which is a combination we can also look in the complementary dimension and see that this is the same as the stuff that is perpendicular to the normal vector we can describe it either by describing what it is or by describing what it's not like here the black is describing what it is how you build it you go to some place on the plane with our position vector and then you move in some combination of the black and what it is or we can describe it by saying go to some place on the plane and move in any direction that isn't that way I can describe this floor as saying I can get to places on this floor by walking in combinations that way and that way from where I'm standing or I can get to any place on this floor by doing whatever I can do as long as I don't jump and they both describe this floor if I jump in any direction that's forbidden this is all the stuff that is perpendicular to the normal and now we can see these are sort of the same we can find some unity in these two things so in particular we can describe a plane in this way but we can also describe it as everything with dot product with the normal is zero shifted by some given point so that p let me give this position guide some let's call this v0 sorry v0 is my position from the origin to some place on the plane and so this will really be vectors v so that these are going to be vectors of the form here so these guys v so that when I subtract off v0 then I'm left with something which is normal so this is I guess I called that a so I have this vector a here and so I need that v minus a this vector which is the point in the plane has to be perpendicular to my normal vector so if I'm given a point in the plane mainly the vector that sticks to the plane somehow if I take a let me draw this picture because I'm sort of trapped so here is some vector a that points somewhere in the plane and then I can describe the rest of the plane as a combination of these two black vectors with a or I can say well if I take any vector that lands in the plane like this one so this is some vector v points at some point in the plane and now I look at v minus a which is now a vector laying in that plane v minus a is not necessarily laying in that plane it's parallel to something in here v's so oh you said v's I mean it's parallel to the plane don't check this in the plane by multiplying with the normal vector well that's one way it has to be true and it can't be true for anything else so this is a combination or you can view it as a defining property so let me do an example with actual numbers we know v's in the plane already I don't know what v is v is all points that live in the plane why would we not have the n to see it okay so let me so here I'm going to give you a normal vector which is 1, 2, 3 there's a normal vector in 3 space that's a terrible one to draw how about 1, 1 that one's easier to draw one way this way one way this way and up 2 that's my normal vector now of course I can pick this up and drag it around but now I have a plane put it up there this is also because vectors we can pick up and drag around that's displacement so this is the vector 1, 1, 2 and this is the plane consisting of all vectors which are perpendicular to 1, 1, 2 but there are lots of vectors that are perpendicular to 1, 1, 2 because I can slide this plane up and down 1, 1, 2 so I have one other condition that my plane contains the point I don't know so let's this plane well then that picture is wrong 1, 2, 3 0, 3, 0 so this cuts through at the point 0, 3, 0 so my plane contains 0, 3, 0 so now let's describe that plane so one way I can describe this plane is to find a vector this way and a vector that way you said that B minus A necessarily lies in the plane this is A and I get that A lies in the plane but I feel like you multiply B minus A dot N you do that in order to see if it equals 0 if I know B but if I don't know B there's not really a point to that if you know that B lies in the plane you're creating a function whose only solution are the points that lie in the plane so here's some vector B who lives in the plane ok and I'm saying that my plane is described by all vectors B which is a variable just like when you write y equals 3x plus 2 you're saying tell me x I can tell you how to calculate B here I'm saying I mean y here I'm saying N and tell me A and we can calculate B by saying I'll give a better way to calculate it in a minute it's any vector B so this is a variable so that let me finish B minus 0 3 0 1 1 2 equals 0 this is defining B if you want you can call B x, y, and z and this will give me equations for x, y, and z that must hold true so my B is x, y, z and I'm going to switch back to the last and so now I can write equations in terms of x, y, and z that will only be satisfied by those triples x, y, z namely I have to have x minus 0 y minus 3 z minus 0 dotted with 1, 1, 2 has to equal 0 in other words x plus y minus 3 plus 2z has to equal 0 now I have a slightly more familiar non-vector description of the plane in terms of triples x, y, and z that satisfied this relationship I just gave it to you in terms so this is a restriction again this equation is saying consider all possible x, y, and z throw away the ones for which this is false this is constructed this is saying so I can also describe this if I have my other two vectors saying here's how you can build anybody who lives in a plane start with my vector v take my two vectors u and w and add together all combinations of the u and w so one is a restriction and the other one is a recipe for building contexts just like for a line you can usually write y equals some slope times x plus something else this is a restriction but if we write it this way go to this point and then move in that direction it's a recipe for constructing this one is sort of inside and this one is describing what is not outside inside they give us the same thing and they're both useful in two different contexts depending on the situation so in general the form of the equations of a plane v space are going to be of the form x minus I don't know let's write it plus vy plus cz equals and all I did is I took that other equation over there and I gathered together all the coefficients and I threw them on the other side and we can divide through by one of these if you want you can divide out one so I don't really have four things as long as a is in zero one of them isn't zero so you can pick one of them and divide it out it's a three dimensional thing so we expect to have three degrees of freedom to describe it but if one of them is zero I can't divide by one just like with a line usually you could say that a line in R2 is ax plus by c and you can get rid of one of these as long as it's not zero then you get rid of the other okay so now let's use that notion to see how we figure out I don't know so we can use these kinds of ideas to figure out things like let's do something simple first so say I have some line at some point so here's my line now then maybe it's in three space maybe it's not and here's my point B and I want to figure out this distance let's say this is my arm I don't know q plus t times v let's think of it that way rather than the normal thing and how would I figure out the distance I already did this actually on the homework problem I just did it a minute ago but make a line perpendicular so here let's do this as a problem so this is the point two, three, four and this is the line one, one, one it's a vector plus t times one, zero, one okay so what do we do to calculate the distance two x minus two, y minus three c minus four and then you would know this with a vector on the line and that would be equal to zero and that would have to be equal to zero wait a minute so you're saying take some point so where does this guy it's not even over that's the vector that would be the problem so where is x, y, z in these pictures x, y, z would be any other point on the baseline well so since you did two, three, four and that's saying it's perpendicular this, where's my origin one, one, one let's say my origin is somewhere here you're saying it's perpendicular to this vector oh I see you're saying this vector okay so this vector here is the vector of x minus two okay that should work did you have a different one there's a dash that just low one plus t, low one this exists on both lines so the line should be low one plus t, low one minus two, three, four that should be the another vector and it's dot dot, low one okay one, one, one plus t let's just do it this way t, zero, t, p, f minus two, three, four so that will be a vector along this line yeah the whole dot, the whole thing dot, multiple, follow one it's zero this should be the same problem this should be the same another way you can do it in this way you can go with three of one numbers there should be well there's even another way I mean they're all the same but not really the same if you go one, one, one from both the line and the point so that the line ends up going through the origin okay and then you take the vector to the new point and find its projection on the line and you subtract the projection from the vector right so these are all sort of equivalent but maybe it's not obvious that they're equivalent so let me say what he said this is sort of the way I think of it these are all fine I think this is the most computationally effective way but what really matters is that you can conceptualize the problem in a way that enables you to solve it and remember how to solve it so what he suggested let me just draw the picture again so here's one, one, one here's the origin here's my one, zero, one obviously this is not the scale and here's my point two, three, four and I think what you suggested please tell me if I'm wrong is look at the vector point one, one, one two, two, three, four and project it on to the vector one, oh, one and that gives you a little subtract to the way it's exactly the problem that we started with last week so that is if I take two minus one three minus one four minus one and dot it with one, oh, one and then divide it by the length of one, oh, one that will give me this red length and I want to turn it into a vector in the one, oh, one direction so that'll be this is in the direction one, oh, one divided by the length of one, oh, one which is the square root of two so this is really the two on the bottom that's the red vector and now I can just compute the green vector minus the red vector to get the vector this length I want length of green these are all if you think about them long enough they're all the same but I think it's worth realizing that there are three seemingly different or four or 12, they're seemingly different ways of attacking this same problem and they're all okay there's nothing wrong with this there's nothing wrong with this there's nothing wrong with any of them think about the problem, conceptualize it and then it should work I'm not going to finish this calculation because I'll probably mess it up and I'll be embarrassed and cry okay what about the situation I have some claim and I want to find the distance what would I do, same as that I just want to project this guy down here some vector that's a combination then calculate this I guess I'm going to just draw a red and a black yeah draw a red and a black so draw a red and a black okay, so here's a red and a black no, on the plane okay, fun so suppose I have my plane so the question is whether my plane is described in terms of a normal or in terms of not an normal in terms of a pair of vectors yeah, so here I have some, let's call it okay, so there's some line okay, and you draw a red line does it have to contain this point? no, it doesn't yeah, there's my line okay okay, use the line as a you draw a circle across the line and the point should be okay, that I didn't understand okay, can I just show okay, basically this is a line yes, and this is the midpoint and draw a circle just across the line and you can prove that this is actually a particular thing yeah, okay so that's the line we need okay okay yeah, I guess so wait, you could also cast this in terms of vectors by pointing this guy acting it there and projecting it here and then adding them up so again, there are a bunch of ways you can do this let me, how are we doing on time? I want to move along because yeah, I want to get something so all of these kinds of problems really come down to thinking about dot products and projections and things like that but one thing that is useful thinking about these planes is if I'm given two vectors that live in the plane oh, this is the way I write this the way I spell suppose I'm going to keep it out at the beginning how I spell suppose so suppose the plane is a plus t u plus s v and I want to turn it into I want to find the other form so that is I want to find a vector in so that u is m n dot h v both together now certainly one thing I can do is I can write n as x y z and I can write n as x y z and this will give me an equation in three variables and this will give me an equation in three variables it's all those two linear equations simultaneously with an equation in one variable which will tell me n but it will tell me a direction and I can choose any value so I could do that but it's not very efficient is this clear that I could do that so I could just take x y z dotted with u x y z dotted with v set them both to zero simultaneously get rid of one of the variables and I'll get something like z equals well I'll get some relation in a moment I can do that I can get z in terms of f of x and y which will be a linear relation and then I take any one value and away I go I don't really want to do that because that's sucky so there's a very useful thing in three dimensions doesn't work in higher dimensions there are analogs of it in higher dimensions it is also so we have this inner product or this dot product we have another way to combine vectors called the cross plot it has some horrible formula that I can never remember so we write this as u cross v and it works kind of like a product except it is not commutative which comes down to doing this so yeah I could do that so I'm just letting you know because I can't remember the formula I can guess at the formula and I'll probably get a side note because it's going to be well here I wrote it in my notes it is so u2 so u is u2 u2 u is u1 u2 u3 and v is maybe I should call it w so that it doesn't look like a u w1 w2 w3 and then the formula is u2 w3 minus u2 yes comma don't memorize this because you blow it just like me the one that doesn't involve the two and who comes first still you u3 w1 minus u1 w3 and then the last one must be that's not right we did that one u1 w2 u1 w2 minus u2 w1 so that's the definition it's horrible but as he said how many of you have seen this before how many of you have not seen this before so if you took physics you saw this before and if you didn't maybe anyway another way, an easier way to remember this relies on knowing determinants a little bit but mnemonically it's the determinant of so I introduced this notation a little bit but I didn't use it we put at the top we put u's components here it's here I will talk more about the determinant in a bit and then we calculate the determinant of this matrix which means that we take the products this way so we take this product gives me in the i component u2 so let me do it separately so I can remember it we multiply everything in the down diagonal this way so let's write it this way i2 so I multiply these ways and then I subtract off multiplying so I would get i u2 u3 u3 w1 plus k u1 w2 and then I subtract off i going the other way here here so i u3 w2 j u1 u2 u3 so once you get used to it it's not so bad so let me do an example with numbers so suppose I want a vector perpendicular to 1, 2, 3 and 1, 0, negative 1 so then I would just write i jk 1, 2, 3 1, 0, negative 1 and then I would calculate the determinant which is going to be in the first component which is i 2 times negative 1 is negative 2 and then I subtract off 3 times 0 in the j component I take 1 times negative 1 which is negative 1 and I subtract off do they get the sign wrong here? no I subtract off 3 times 1 and then k component I take 1 times 0 2 times 1 is usually 2 so this would be negative 2 negative 4 negative 2 and we should check that this really works negative 2 times that's negative 2 plus 2 so it's certainly perpendicular with this one if I take the dot product and if I take the dot product with this one I get negative 2 so I get negative 2 minus 8 is minus 10 and then I have a sign wrong somewhere do I mill something? yeah cause I'm supposed to swap the sign there okay so then I get minus 2 minus 3 is and minus 6 is 8 and 2 times 4 is 0 so that works and I got the sign wrong here too these are both minus yeah cause when I take a determinant I'm supposed to alternate signs so sorry I messed that up so when you take the determinant every other row is plus minus plus minus we will talk more about determinants when we talk more about matrices and things like that sorry I screwed up but anyway you can also just solve these equations in a long complicated process so what is the point of this? well it enables us to given a pair of vectors find a vector perpendicular to it quite readily and so we can now switch back and forth from this form to the form involved in the normal quite easily because we can find the normal vector by taking the dot product now one thing I mean the cross product one property of the cross product though the eraser is that it is not commutative that is v cross w is not the same as w cross v it's anti commutative swap the order of the sign changes that corresponds to in this matrix formula swapping the position of these two vectors these two rows which changes the sign geometrically if I have a vector v here and a vector w here then the vector v cross w will be up here and w cross v will be down there so this is called the right hand rule I mentioned this before put my hand along v curl it towards w my thumb points in the v cross w direction put my hand on the v vector curl my fingers towards put my right hand if you're doing your left hand it won't work put my right hand on v curl it towards w my thumb points in v cross w if I do it with my left hand it hurts too much to try put my hand in curl my thumb points for long how do you generalize the vector for determinants and matrices the cross product really only works in three dimensions you can generalize this kind of thing but so it only works as long as the numbers just come out so that's like assuming we're talking about actual space so well so I mean we can let's say no you had a question I was wondering well so kind so not really so here this is in R3 we find quaternions that have a non-communitive multiplication but that's a four dimensional space it's kind of like the generalization of but it's not really a cross product but you have something similar where if you put the two the other one out right here if we think about the I cross J is K where this is the unit vector along X this is the unit vector along Y this is the unit vector along Z and J cross A is I so they sort of go around in a circle and for the quaternions we can also totally one so if we have one I, J, and K so then we have four mutually perpendicular dimensions and we can make an algebraic structure out of this sometimes called a Hamiltonian in the same way that there's an algebra for two vectors called the complexes there's an algebra for four vectors called the Hamiltonians and there the multiplication operator is not communication and you can even go to octonians where you also use association so you can do sort of an algebraic structure with one number, with two numbers which is complexes with four numbers the Hamiltonians and the octonians but as you go up more and more you keep losing more and more of the standard operations that you expect to work so if you go to octonians you don't have association but that's not in this class are we okay with that? so one thing that this is useful for is for us to construct perpendicular vectors which can be handy when we want to project stuff for instances and find equations of planes and all sorts of stuff like that it also which has to do it has to do with this determinant properties it also notice that I didn't use anything about the length of v cross w I just said v cross w it goes that way and this length could be quite long so another sometimes useful thing if I have three vectors so those are supposed to be vectors in three space they are not any of them perpendicular you can notice that they sort of define a prism in a box my picture is terrible something is wrong here that was too long so I have box like that whose sides are u, v and w and certainly if I calculate v cross w this gives me something in who is sort of in the direction that v is in except v sort of over and we could project this guy to that guy using a dot product if you mess around with this well then you would see if I take this angle here then this length is u cosine theta this length if I calculate u dot is v cross w that's a number and it corresponds to the volume it's not completely obvious but if you sit down and calculate it out you can see that it doesn't work so this is a way to calculate volumes of boxes and things like that where the sides are not perpendicular so so what do we have we have five links left so I could do some more all I wanted to say from chapter one I can go through that calculation but I really don't feel like it and I'll probably screw it up and it's not, I mean this is not like a huge, magically important property so this is sometimes called by the way the triple which seems like it should be I don't know why it's called that well because it's a triple product and it gives you a scalar it's zero well it'll be zero if any one of them lives in the same place yeah the order doesn't matter it's just if you get a negative positive for example so if I exchange the roles of v and w then you can switch if I twist them around I'll get the same out so this is the volume and the order doesn't matter which if you think about it it's the same box and I'm just projecting on the same other side you know just like when you have a triangle and you want to compute its area it's the base times the height and it doesn't matter if you choose as the base you get the same number it's exactly the same property okay so I mean I could do a few classes we've talked about lines and planes and stuff like that but there is a relationship here also and I want to sort of get this notion of determinants and things like that there is a relationship with equations linear equations let me hold half of the word linear equations I have some system of equations like that then I have three equations and three unknowns so I would either have no solution or I will have a whole plane or line or something of infinitely many solutions or I'll have exactly one solution and probably many of you have encountered these kinds of things and maybe you even know something about writing this in terms of matrices who does this not look familiar to a few of you okay it's alright so and there is a close relationship these are actually vectors and transformations of vectors if we think of this x, y, z as a vector which could be a three vector it could be a twelve vector I don't care and we think of this as a vector and this is a function that tells me I put in a vector and out comes another vector and that vector will involve x's and y's and z's here is that vector and when I set an equation I'll get another vector so we can recast this problem about systems of equations in terms of vectors and transformations on the vectors or in the context of matrix times a vector and one of the main points, one of the main uses of this idea of linear algebra is seeing that these two questions thinking about vectors as describing geometric objects in space and thinking about vectors as describing systems of equations are the same question and we need to do this kind of thing of that kind of thing to do calculus in higher dimensions but a lot of times this kind of jump falls out of it too and so what I'm going to do on Monday is talk about the relationship at least briefly between messing with vectors and transformations of vectors and systems of equations are really the same question and really when people realize that these were the same question is why this field of linear algebra was developed because it's the same thing just like thinking about lines and thinking about linear functions this is how we stop there so I will put another homework assignment sorry the one that we do today and the next one will be within 24 hours and maybe within one can I forget about of course I can forget about it it's still due next week but if I can forget about it