the duality between quaternions and vectors

Transcript

today we're going to look at a nice connection between the querian and a lot of like vector identities in R3 but let's just start by recalling what the querian are so as a set they're linear combinations of the number one this Vector called I this Vector called J and this Vector called K or maybe you could just think about those as imaginary quarians and not vectors but this does form a vector space and this is a real Vector space so a b c and d are real numbers and then if you do multiplication between two things from this set you have to keep a couple things in mind first of all I S J2 and K2 are all -1 so you've got three different or actually six different if you think about positives and negatives roots of Nega one or square roots of negative one then if you multiply I with J you get K whereas if you do the opposite and multiply J * I you get Negative K so it's not commutative furthermore J * K is I whereas K * J is negative I and K * I is J whereas I * K is negative J and maybe I'll just also point out that our multiplication here is associative and you can actually check that from from this given information you don't need anything else but I won't do that so like I said we're going to compare this with operations on vectors in R3 so let's just really quickly recall the cross product here recall the idea of the cross product is it produces a vector that is perpendicular or orthogonal to the two vectors that you are taking the cross product of and then we have the dot product okay so let's maybe start by taking two elements of the querian I'm going to write them like this so maybe U1 plus X1 in the I Direction Plus y1 in the J Direction plus Z1 in the K Direction and then I'm going to multiply that with the Quan uh U2 and then you might guess what's going on here plus X2 i+ Y2 J + Z2 K and If I multiply this I'm going to get another linear combination of the number one i j and k so really I just need to think about the contributing factors for all of those terms so let's look at the real term first the one that's not attached to i j or k and let's observe that we get that real term from this U1 * U2 also from this X1 I times this X2 J maybe I'll do a double underline just to show that those are matched and then this y1 J times this Y2 J and finally this uh Z1 K and this Z2 K now of course when I multiplies with I you get a Nega one so we're going to have a little bit of a difference of a sign here okay so anyway the real part which I'll put in these magenta parentheses just so that we know where it came from will be U1 * U2 minus X1 * X2 and then plus y1 * Y2 + Z1 * Z2 where observe what I did here is I just grouped all of those together so I could have minus all of them again this really is attached to an i s this term right here the Y's are attached to a j s and then this next term the Z1 and the Z2 are attached to a k squ but we know that they those all square to the number one and then next after that we're going to have something attached to I so let's see what that will be so let's see we can get that by doing uh U1 times an X2 I and then we can also get that from doing a couple of other things so an X1 I with a U2 so that's another way we could do let's see uh y1 J multiplied into a Z1 K and then finally a Z1 K multiplied into a Y2 J now after multiplying that out we of course need to keep in mind that the sign will differ based off of let's see this rule right here so if J is on the left and K is on the right we pick up an i if it's opposite we pick up a negative I so let's just quickly write out what we get from that we'll have a U1 * an X2 and then plus let's see a U2 * an X1 and then after that we'll have and then plus a y1 and a Z2 and then minus a Y2 and a Z1 and I'm actually going to group these last two terms and we'll see why that is in just a second so let's look at those and this y1 Z2 minus Y2 Z1 observe that looks like this first entry and the cross product okay so that's pretty interesting okay so let's get the rest of this product on the board I'll let you guys work out all the details if you need to okay so there we've got that on the board and now well let's take a closer look at this and maybe in order to really see what's going on here let's pretend that the so-called real part of these starting quarians is equal to zero and so that means all of the U's are zero so I'll just cross out all of the U's so that's going to be gone this term U2 is going to be gone then means this U1 * U2 will be gone and then all of these first chunks of the rest of this will also be gone oh and I noticed that I forgot this I this J and this K here okay so let's get rid of all of those and now let's observe what we have here so we've got this x1i this y1 J this Z1 K and then similarly with the subscript twos over here furthermore notice that this sum right here that's exactly the dotproduct of those wouldbe vectors whereas this term this term and this term are the corresponding entries of our cross product of these vectors over here so there is definitely some sort of connection going on here okay so let's pick up on the next board with a reframing this multiplication in a way that will help help us towards our goal which like I said at the beginning is to come up with some Vector identities okay so now that we've seen a bit of a parallel between the querian product and the dot product and the cross product let's make it a little bit more succinct or more efficient in terms of notation so we can find our Vector identities so let's say we've got three querian we have q1 Q2 and Q3 and we're kind of simultaneously thinking of them as vectors in R3 and querian so I'll write this as a vector B vector and C vector and we have A1 I plus a2j plus A3 K and then similarly for B1 B2 and B3 and I'll just put like a note here as to how this product works and I'll just do the product with q1 and Q2 and as we saw on the last board the real part of this product ends up being negative the dot product between these vectors so a do B and then the imaginary part ends up being well the cross product of these vectors and so if it's not clear the real part is everything that's not attached to the IJ and K so in other words just the free real number and then the imaginary part actually has three components okay so that's going to be plus like I said the cross product of A and B where we're kind of bending our mind to simultaneously think of these as querian and three vectors like I said before and then maybe I'll go over here and just recall the fact that the multiplication is associative like I mentioned before so that means q1 * Q2 * Q3 is equal to q1 Q2 and then times Q3 and that's sort of the whole point here okay so let's start with this left hand side and see what we get so we have q1 * Q2 * Q3 so that ends up being let's see q1 times now taking the querian product of Q2 and Q3 in up being what so it's going to be negative the dotproduct of B and C plus the cross product of B and C and then well really quickly let's recall that this part right here is the real part of this product of q1 and Q3 or Q2 and Q3 whereas this thing right here this cross product is the imaginary part of Q2 * Q3 so doing this produ product is a little bit tricky so I'm going to do it in stages so let's take this q1 keeping in mind that it is all imaginary and we'll multiply it through to this real part so that's going to give us negative the dotproduct of B and C in the a direction recall that b do c is just a number so that just scales this Vector a and then after that we'll have what I'll call the querian q1 multiplied into B Cross C and I'm going to write it like that with this querian multiplication because the querian multiplication works like this thing that I have written as a note up here okay so now let's bring this down and we have negative the dotproduct of B and C in the a direction and then q1 multiplied into that is going to have two components so the first component will be this a dotted into B Cross C and then this next component will be related to this right here so that'll be plus a cross the cross product of B and C okay so let's note that now we've got a real part which is only right here so that's kind of interesting and we have two things that are related to the imaginary part this term right here and then this term over here okay so let's bring that final product up and then we'll do the multiplication for the other Association okay so this is where we ended up on the last we decomposed back into real parts and imaginary parts and we can see that the real parts and imaginary Parts because well the imaginary parts are connected to vectors whereas the real parts are connected to numbers here we have a dotproduct of a with B Cross C which gives us a number okay just like I said up here I'll use the same color coding that this term right here is the real part of the product of these two querian whereas this cross product of A and B is the imaginary part of the product of those two quarians okay so now let's get to the product in the other order so we have q1 Q2 multiplied into Q3 so that's not too hard to start because we already have the product of q1 and Q2 up here from our exploration on the last board so that's going to be negative a do B plus A cross B and then multiply it into the querian Q3 and then we're going to view that querian two different ways well as the querian and also as like kind of a vector so this A.B is just a real number so that means that's going to scale the vector that Q3 is this C so we'll have minus a dob in the C Direction and then multiplying A cross B with Q3 uses this definition of well imaginary querian multiplication that we built earlier just with things renamed so let's see It'll be Min - A cross B and then that's going to be crossed into C and then that's going to be dotted into C and then we have the cross product in this case it'll be plus A cross B and then cross with c and now we have to look at what the real parts and the imaginary parts are and I'll use the same color coding so this down here is a real part that I have in yellow because well it's a DOT product um and that gives us well a real number in terms of our querian multiplication and then the rest of this will be our imaginary part so if we have this and we have this but again since quater and multiplication is associative these two are equal and thus these two lines that have the yellow and blue adornments are also equal but that means their real parts are equal and their imaginary parts are also equal so let's see that means from this real part we get the vector identity which says a dot b Cross C is the same thing as a cross B do c so that's a well-known Vector identity and then extracting the imaginary parts we get another well-known Vector identity and I'm going to move some things around just to put it in its most classical version and that is a cross B Cross C minus A cross B crossed into C is the same thing as B do c in the a direction and then minus a do B in the C Direction and there you have it we've gotten these two nice classical Vector identities using querian multiplication and that's a good place to stop