Geometric Algebra in 3D - Fundamentals

Transcript

this is going to be a fun video because we're going to finally start talking about geometric algebra in three dimensions what we're going to do is first review what we've been able to set up in G2 the geometric algebra of two dimensions and then we're going to finally set up G3 the geometric algebra of three dimensions we're going to add a few more mathematical objects and if I've been explained this stuff clearly it shouldn't feel like you've actually learned too much in this video let me quickly review what we did to set up G2 what we did was start off with some vectors namely two vectors we had one vector which I drew as this horizontally directed Vector called E1 this is of length one and we had another Vector orthogonal to it also of length one called E2 so two vectors in our basis for G2 E1 and E2 I also call these sorts of objects the vectors grade one objects because they're formed by taking one vector at a time we can go down one wrong on the lab to consider the scalers which are formed by taking no vectors and I represent these as just numbers like one or five or -10 geometrically we designate those by points and again these are the grade zero objects and now we Ascend to the highest rung in the ladder to consider the grade two objects and what were these these were the B vectors how do we form the B vectors or how did I motivate the by vectors we were considering the wedge product between E1 and E2 that is E1 wedge E2 which we later found to be just equal to the geometric product E1 * E2 so let me write that there E1 E2 and remember what the geometric interpretation of this was we have E1 then we have E2 there as sides of a parallelogram complete the parallelogram and then we put that circulation symbol in there to denote the orientation of this area element so this is E1 wedge E2 or E1 E2 now remember what this is so this is an abstract area element so this object this by Vector contains information about the area and the orientation of the area using the language of linear algebra we could say that G2 the geometric algebra of R2 is a four-dimensional abstract Vector space because we have four elements in the bases set we have one scaler two vectors and one bi Vector to make up the four-dimensional vector space now in addition to defining what the mathematical objects are we want to do some operations and the principal operation that we're concerned with in our talk of geometric algebra is the geometric product and abstractly this has a lot of nice features just to remind you what those are it's associative so if I have three vectors UV and w UV * W on the right is equal to VW * U on the left so it's associative it also obeys the distributive law so U * the sum of V plus W is equal to UV plus UW and it has a another nice property which is that if you square any Vector let's say U you square the vector under under the G mat product that's equal to just U * U that's equal to a scaler which scaler is it it's the square length of the vector so these are some of the nice properties of the geometric product that I just want to remind you of besides the geometric product we had a couple other very useful operations that we could Define we were talking about the dot product quite a bit remember what this dot product is this is an operation that takes in two vectors and it outputs a scalar and we write this as U Dov so inputs two vectors to grade one objects and outputs a grade zero object so it's grade lowering and UV can also be written as2 UV plus v so in terms of the the philosophy here we could think of the geometric product as the the principal operation whereas the do product is one of the subsidiary operations because it can be defined in terms of the geometric product besides the dot product we also had the wedge product this is the other pretty useful operation now the do product is grade lowering the wedge product is grade increasing so remember when we take in two vectors U wedge V both grade one objects we output a b vector and like the dipr this can be defin in term of the geometric product as well this time it's 12 UV minus Vu and hopefully you can see that when you add this to this you can write the geometric product between two vect v s UV as the sum of the dot product plus the wedge product and this is actually how I motivated the whole Talk of the geometric product but it's probably more proper to view the geometric product as the underlying operation in this algebra and the dot and wedge products is subsidiary operations another last note on these Dot and wedge products we also discovered that when we have two vectors which are orthogonal that the geometric product an and that's readily Apparent from this formula here let's say u and v are orthogonal that means your do prodct go is zero which means UV plus Vu must be equal to zero so that implies that UV is the opposite of Vu and correspondingly we had a nice property when the wedge product goes to zero that happens when two vectors are in the same direction so this going to zero implies that UV minus V has got to be equal to zero which implies that UV is equal to VU the geometric product commutes when the two vectors are in the same direction in general if you had some two arbitrary vectors it's going to be neither of these two cases let me remind you of a couple important computations let's consider the square of E1 remember the square of a vector in the geometric product is the length squared the length of E1 was one so E1 squares to one for the same reason E2 also squares to plus one so that's the first type computation i' like you to recall the second one is what happens when we multiply E1 and E2 remember that the geometric product has a do product part and a wedge product part when you multiply two vectors together in this case the vectors are orthogonal so the dot product part disappears so this is just equal to E1 wedge E2 also remember with the wedge product we can flip the order of these two just to have to stick in a minus sign so this is equal to minus E2 wedge E1 E2 wedge E1 just as the wedge product is equal to geometric product here this wedge product is equal to the geometric product E2 E1 to summarize we have that E1 E2 is equal to minus E2 E1 so when you switch the order of two vectors with uh different numbers You' got to sticking a minus sign and using these two computations we can show that E1 E2 squares to minus one and just to review why that is we have E1 E2 * E1 E2 we're going to switch the order of those two sticking in the minus sign it's going to give us minus E1 2 e22 E1 and E2 both squared of plus one so we're left with a minus one let's now shift our Focus to three dimensions so we're going to set up G3 now we're going to play a very similar game we're going to start off with the vectors this time we're going to have three vectors instead of two vectors we're going to have an E1 an E2 and an E3 and now I'm going to draw these now in three dimensions now so let me draw an E1 as the vector that normally points along what you call the x- axis so that's E1 orthogonal to that also of length one just like E1 is going to be I have E2 going to draw this along the the y- axis and then finally still orthogonal to both of those I have my third Vector pointing along the Z Direction called E3 so just three vectors in three dimensions again these are going to be the grade one objects we have three of these this time we descend to the scalers this is going to be the same thing as before we have one scaler just numbers we represent these as points and these are the grade zeros now to set up the next highest grade the grade twos we're going to play a similar game as before we're going to generate these objects by taking wedge products so the first wedge product I'm going to take is E1 wedge E2 which is still going to be equal to E1 * E2 so the first B Vector is going to be E1 E2 so that's the first combination of two vectors I'm going to take from these three vectors E1 2 so what does that look like it's going to be E1 y G2 so let me attempt to draw this in down here I have my E1 I have my E2 going to draw in that parallelogram there's E1 and E2 that's going to be that way so I'm going to draw the circulation symbol that way so there's your B Vector E1 E2 sitting in three dimensions now so that's one possible by Vector the next possible by Vector I'm going to to consider is wedging E2 with E3 which is also going to be equal to E2 E3 let me draw that in so I have my E2 pointing along the y direction then I have my E3 pointing along Z Direction let me complete the parallelogram now I have E2 E3 so I'm going to draw the circulation symbol that way or the orientation symbol that way and there's the B Vector E2 E3 now sitting in three dimensions let me also point out that the these B vectors are sitting in what you would call here the XY plane here what you call the YZ plane the last combination of two vectors I can choose from this three is going to be E1 and E3 now this time I'm going to actually write E3 E1 going to wedge it in the order E3 E1 even though it's arbitrary I'm just going to do this for convenience so let me draw that into so I have uh E3 first pointing in the Z Direction then I have E1 let me complete that parallelogram I have E3 E1 so the circulation symbol is got to be drawn like that so there's your E3 E1 by a factor saying the three dimensions and this is going to be on the XZ plane what you normally call the XZ plane so we3 so we see that in G3 we have three B vectors instead of the singular B Vector in G2 so let's summarize we have one scaler three vectors and three B vectors from the algebraic point of view one property that we want of this basis set for G3 is that it be closed under the gometric product that is if we take the product of any two of these elements that we have so far we get another element that's already in the set now for the most part that's not a problem if you consider the product of the scaler times any one of these you'll just get that same grade back if you consider the product of two vectors for example let's say you have E1 * E1 you get a scaler back if you have let's say E1 * E2 or E1 * E3 you get a B Vector now let's consider the other case where you have a b Vector time a vector let's see if that's closed now if you multiply E1 * E1 1 E2 that's equal to E1 s * E2 which is just equal to E2 we already have that but let's consider this other case where we have E1 E2 * E3 that's something that's not on the list yet so we have E1 E2 * E3 that's not on the list so that's actually our last basis element in G3 E1 E2 E3 and this is going to be called as you might expect a tri vector and this is that last grade three element because this is going to be formed by taking the wedge product of all three of the basis vectors at the same time E1 wedge E2 wedge E3 compare that to the B vectors here where you take two of the three vectors in the bases and wedge them together the tri Vector we're wedging all three together and we're actually going to have but one Tri Vector in G3 so we can see we have actually all the basis elements we need we have one scaler three Tri three vectors three bi vectors and one Tri Vector so again as an abstract Vector space the dimension of G3 is going to be eight dimensional because we have eight basis elements we're going to explore that Tri Vector in just a moment but I'd like to call your attention to something perhaps more important which is that when it comes doing calculations with the geometric product there there's nothing you really don't know how to do already let's review we found that the vectors square of one remember we were saying that E1 squares to one E2 squares one for the exact same reason E3 also squares to one so let me write that all these vectors Square to 1 E12 = e22 = e32 equals 1 for identical reasons that we found that E1 E2 is minus E 2 E1 flipping the order of any of these two will result in you having to put in a minus sign so we found before in G2 that E1 E2 is minus E2 E1 and for the exact same reasons if you go through the algebra E2 E3 is going to be equal to minus E3 E2 and finally we have that third relation that e e 3 E1 is going to be equal to minus E1 E3 hopefully you're seeing conceptually what's going on if you have two vectors and the numbers are different you want to flip the order you got to sticking a minus sign they anti-c commute and finally just as we saw in G2 that single B Vector E1 E2 squares to minus one the same holds for E2 E3 and E3 E1 those two also Square to minus one and let's review why that is uh taking into account these anti-c commutative relations let's try finding E2 E3 squar so it's going to equal to E2 E3 * E2 E3 we use this property here we're going to swap this and this we're going to get minus E2 2 * E3 2 these both Square to one using this up here so we get minus one and you can play the same game for E3 E1 E3 E1 * E3 E1 switch this and this we get minus E3 2 * E1 2 and again that's equal to minus one so let's write out the full statement for the B vectors we have that E1 E2 s is the same as E2 E3 squared and that's the same as E3 E1 squ those are all equal to minus one so you can see there's a nice conceptual unification going on here what you know from G2 is going to nicely transfer over to G3 we've been squaring stuff so let's just find out what this Tri Vector squares to because we already know what the squares of all the other things are so let's consider E1 E2 E3 squared that's equal to E1 E2 E3 times E1 E2 E3 now what am I going to do I'm going to move that E3 two places to the right so I'm going to move it here then here that's two swaps so this going to be minus minus so I'm going to put in two minus signs but those are going to cancel because it's it's an even number I'm going to rewrite this as E1 E2 the E3 is going to get shuffled over to there the E1 E2 are going to move over times E1 E2 then I have E3 E3 or just e32 that's equal to one so this is equal to E1 E2 * E1 E2 which is equal to E1 E2 squared which we've already found is equal to minus one so we conclude that the square of this Tri Vector is also minus one so this is yet another object in geometric algebra that squares to minus one hopefully you can see you can do pretty much any computation you want to now with the geometric product I think you already know how to do it you may not know the significance but you can do the computation anyway for example if you had 5 + 2 * E1 E2 you want to multiply that by 2 plus E1 plus E1 E2 E3 you just follow the rules distribute the stuff out remember that the vector Square to plus one the B Vector Square to minus one the tri Vector squares to minus one when you want to flip the order of these two or these or these or anything in here you've got to STI a minus sign so that's multiplication addition is as you would expect you can combine stuff when you can for example if you had something like 2 E1 E2 and you wanted to add that to let's say five E1 E2 of course we can combine them in this case you won't always be able to combine things for example if you had 2 E1 E2 and you had let's say plus five E3 E2 you can add these bi vectors together and there's actually a geometric interpretation for this but you can't really do much more than this although you could mess around with the factor you could factor out the E2 you could write this as 2 E1 plus 5 E3 Multiplied on the right by E2 but sometimes there's not much more you can do you can buy them when you can and when you can't don't worry about just leave them alone hopefully we have some geometric understanding of what vectors and V vectors are remember the V vectors those are the oriented pieces of area pieces of abstract area they have no particular shape but they do have an amount of area and an orientation to them now let's try to understand what this E1 E2 E3 Tri Vector is now I claim this is simply equal to E1 wedge E2 wedge E3 this is a claim I'll justify in future videos but this is hopefully it's it's it's intuitive I form the highest grade element of G3 by wedging all three vectors together simultaneously now let's try to understand what this statement is E1 Y2 wed E3 let's draw in something that we're probably more familiar with the E1 wedge E2 by a vector there's E1 E2 complete the parallelogram put that in now let's recall what this wedge product is doing it's forming a new geometric figure by taking one figure and extending it along the other so you can imagine this area being developed by taking this E1 vector and sliding it in the E3 Direction so it sweeps out that area so that's how you get a new geometric figure from the product from the wedge product of two lower grade objects so we're going to apply that same idea we're going to take what we've got here this B Vector this patch of area and extend it in the E3 Direction so we're going to extend it in the E3 Direction so what's that going to look like I'm going to attempt to stack these patches of area on top of one another you can imagine this as a kind of a summing of those areas but still extending this geometric figure until we get a new geometric figure and what's this geometric figure going to be when I complete that extension what I'm actually going to get is an oriented piece of volume so I'm going to draw these in by these cubes here even though we should be thinking of these abstractly this is a a mathematical object which has volume information and has the information of the orientation of the volume but it has no particular shape you can imagine as a blob but just for convenience I'll draw these as cubes or parallel pipids just as for the B vectors these are also abstract but for convenience I draw those as parallelograms so this is the main geometric Insight I'd like to to carry away with these Tri vectors is that they generate oriented pieces of volume so we're going just as we're talking about positive areas and negative areas we can talk about positive volumes and negative volumes too and just as when we wedge E1 and E2 the magnitude of the area that we get is one that's pretty obvious because we set these up the E1 E2 to have length one so the product of those two is the enclosed area E1 E2 and E3 all have length one they're all orthogonal so the volume is just a product of all three of those so 1 * 1 * 1 the volume enclosed is one so the magnitude of E1 E2 E3 is always going to be one and by convention we're going to say that this thing the E1 E2 E3 is going to be+ one there are many interesting things about this oriented volume element that we've just generated but let me point out a few things just using this wedge product so we started with E1 wedge E2 wedge E3 let's play around with this a little bit remember that when I switch the order of any two I've got to stick in the minus sign so let me switch the order of these two so I have minus E2 wedge E1 wedge G3 and finally let me switch the order of these two so I have plus E2 wedge E3 y G1 so this is the same thing as this so that means that the geometric figure of this which is this Cube this volume element here is the same as this one so we interpreted this one through extension of the by Vector through E3 to generate this abstract volume element so let's do the same thing with this statement here so this is saying that the wedging of E2 and E3 which is going to be this by Vector here E2 W G3 let me draw the orientation there E2 W G3 is got to get extended in the E1 Direction and that generates the same abstract volume element so hopefully you can see that if you take this face of the cube and you extend it this way in the E1 Direction you also generate this abstract volume element and actually if you play around with this a little bit more you can also show that E3 wedge E1 wedge E2 is the same as this and the same as this which corresponds to taking this by Vector on this face of the cube which is going to have that orientation and extending it in the E2 Direction and of course I could always write these in a more compact fashion I said that E1 E2 E1 W G2 W G3 is equal to E1 E2 E3 this can also be written in the more compact way which is E2 * E3 * E1 now this product here is a geometric product instead of the wedge product and that's also equal to E3 E1 E2 and as I claimed before you already know how to do quite a bit with geometry product in terms of flipping things and putting minus signs in so you can play around this yourself and confirm that these are indeed equal and again you can swap things around as you'd like let's say we had E1 E2 E3 and we swap the first two vectors around we get minus E2 E1 E3 we get can negate both sides and this is saying that E2 E1 E3 is equal to minus E1 E2 E3 so what is this Tri vector or E2 E1 E3 well it's just the opposite of that original volume element that we had so here we're getting some negative volume if we're going to Define these as positive volume we're getting some negative volume here and if you interpret this last statement again through extension this is saying that if you take E2 wedge E1 which is going to be this by Vector here but it's actually going to be the opposite orientation and you extend that along the E3 you're going to actually get the opposite of what we originally generated which was E1 E2 E3 there's one other property of the tri Vector that I'd like you to attempt to discover on your own and the way to do that is just to multiply this Tri vector by each one of these other elements each one of the seven another basis elements and what I'd like you to do is draw what you put in draw the input object whether it's a scalar vector by vector and also draw the thing that's coming out of that product and see if you can discover the general relation between the input and output and therefore what the tri Vector is doing when it multiplies one of these other Seven Elements before I close out this video there are a few pieces of jargon I'd like you to know I've been using these names for the four grade uh the first piece of Jon like you to know is an alter name for these Tri vectors more generally this name is used for the highest grade element in the geometric algebra and that's going to be the pseudo scaler so in G3 the tri vectors are the pseudo scalers and we're going to develop that concept a bit further we just know that these pseudo scales are going to be the highest grade elements of the geometric algebra in that case in the case of G3 that's a tri vector the next highest grade in this case that those are the B vectors those are also called pseudo vectors and again we're going to develop these Concepts in more generality in this case G3 the pseudo vectors are these three by vectors in physics these are sometimes also called by yet another name these are axial vectors so the pseudo prefix is telling me that they're not quite vectors they're false vectors but as I said we're going to develop that concept a bit further in future videos and the vectors there's no other piece of jargon actually there is one uh again in physics in contrast to these axial vectors sometimes people call these true vectors polar vectors so just a few pieces of jargon I'd like you to know the most important ones are the the pseudo scaler for this oriented volume element in G3 and the pseudo vector for these three types of B Vector we're going to be working with as I said earlier if I've been at all successful in this video series I don't think this video really added too much more to your knowledge by the introduction of one more Vector moving from G2 to G3 we add in two more B vectors two more pseudo vectors and then we add in the pseudo scaler the grade three element which is that directed volume element so those are the main Concepts and I think this is going to be a good bit of fun cuz there's a lot of cool physics we're going to be able to do especially with gyroscopic procession which is a video I'm pretty excited to make after all Vsauce did some rotation stuff so I got to steal his idea so hopefully you stay tuned for more videos on geometric algebra and as always leave your angry comments and subscribe to the channel if you enjoyed the content and I thank you for watching