The rotation problem and Hamilton's discovery of quaternions (II) | Famous Math Problems 13b

Transcript

[Music] Hello everyone, I'm Norman Wildberger. This is our  second lecture in this presentation of Hamilton's   discovery of quaternions, essentially solving the  rotation problem—the problem of how to effectively   describe rotations of three-dimensional  space.

In our first video on this topic,   we looked at the planar situation—rotations  in the plane—and the close connection with   the algebra of complex numbers. Today,  we move up to three-dimensional space   and consider rotations about a fixed  point in three-dimensional space.

So,   we're going to talk about rotations,  we're going to get to Euler's theorem   on products of rotations, and we're also going  to introduce dot products and cross products. Now, there are sort of two different ways we can  approach this topic, and they're complementary,   but it's interesting to keep them separate.

One is  a purely geometrical approach where we think about   the physics, the geometry of rotations about  a fixed point, often by working with a model   of a sphere which is centered at the point in  question. This way, rotation of three-dimensional   space can be represented by movements of 

this sphere, and typically, we have an axis,   say something like this—a line that passes through  the center of the sphere and two opposite or antipodal points. One of the main problems  in the subject is to understand how we can   compose or multiply such rotations. Alright, so 

we have this geometrical side of things, but we   also have an analytic side where we introduce  coordinates—typically x, y, and z—and where we   use the linear algebra of matrices, vectors, and  then associated dot products and other algebraic   constructions to help us with the geometry.  So, although these are very closely linked,   one should be aware that there are subtle  differences and perhaps even a little bit   of tension between the two of them.

There are many  advantages to working with the analytic approach,   but there is also a possibility of a little bit of  confusion because when we introduce coordinates,   we are placing distinguished roles on three  particular directions in our space which are   not really inherent in the geometrical picture.  And this introduction of these three distinguished   lines has sort of number-theoretical issues that 

can play a role in connecting these two topics. Alright, so let's start by thinking about  rotations from a geometrical, intuitive,   physical point of view. Alright, so  we're going to be considering these   rotations of three-dimensional space, 

and we're going to consider these as   rigid motions of the entire space which fix a  distinguished fixed point, usually called O,   the origin. And to illustrate this kind of  motion, we're going to often employ the sphere   whose center is the fixed point O. So instead of 

worrying about how the entire space moves, we're   just going to worry about how the points on this  sphere move, because that's easier to represent. Now, the distinguishing property of a rotation is  that it preserves the separation of points on the   sphere. So, for example, if we have two points 

separated by that amount, then if we rotate,   the two points that we're going to get are  still going to be separated by the same amount,   and that's independent of how we actually  decide to measure the separation. So in fact,   there are lots of different possible ways of  talking about how far two points are separated   on the surface of the sphere, but no matter 

which way you choose, that's going to be one   of the properties of rotation—that it preserves  separation. Another important property is that   it preserves orientation. So that means that, for  example, if you had a little circle on the sphere,   and say you had a positive orientation of that 

circle, maybe a counterclockwise orientation,   then if you're going to rotate the sphere,  the image is still going to have that same   counterclockwise rotation. That's as opposed  to a reflection—a reflection in the plane going   through the origin, which has the property  that it reverses orientation.

So reflections   are also an important part of the story,  and we'll come to them a little bit later. Alright, so a rotation is a particular kind of   transformation which has a fixed axis.  So there are two opposite points, or antipodal points. A stumbling block for 

understanding many aspects of three-dimensional   geometry, and I know it's very commonly assumed  that rotations and angles are intimately linked,   but in fact, theoretically, it's not so,  and there are many advantages in avoiding   angles and their essentially transcendental  nature. So to make that point another way,   it's important to realize that a rotation 

is really a transformation that starts in   a certain position and ends in a certain  position, and the actual motion from the   starting position to the final position is  not part of the rotation. So, for example,   let's put the axis up and down like this, and  let's consider the rotation that—in fact, maybe   I'll position things so that this blue line here 

is the equator, and there's an axis something like   that. Alright, so a rotation is something that  takes the sphere in this position and sends it to this position. Alright, I've deliberately moved  the sphere so you didn't see how I went from this   position to this position.

So perhaps I went like  this, or perhaps I went like this, something like   that. But the point is that the rotation itself  is really determined by the initial position   and the final position, and not the actual  motion from the initial to the final position.

So on our diagrams, we're going to have pictures  like this: our sphere represented by this little   picture here, and here maybe is the axis of  the rotation with fixed points like the North   Pole and the South Pole, say A and A-bar,  and we have some rotation which sends, say,   that point B to that point B-prime, that point  C to that point C-prime, that point D on the   equator to that point D-prime.

But we'll try to  avoid drawing arrows necessarily indicating that   we have to actually move in this direction as  opposed to going the other way around. The key   point is that this point is sent to this point,  and how it actually gets there doesn't concern us.

It's useful to mention that angles come from  astronomy, and in astronomy, there is a very   good reason why angles are important, because  the Earth is rotating uniformly on its axis once   a day. So everything that we observe in terms of  the night sky is moving around us at a fixed rate   because of this uniform motion.

So in astronomy,  angles are completely unavoidable and play a very   important role. Surprisingly though, when we're  studying the geometry of three-dimensional space,   and in particular rotations and reflections  and other motions, we don't need to use angles,   and often using angles makes life more 

difficult for us. That's going to be   an important idea that will even inform our  discussion of quaternions in our next lecture. So one of the main problems in the subject  is to describe what happens when we combine   rotations.

So in other words, we  have one rotation about one axis,   and then perhaps followed by another rotation  about another axis, and we want to consider   the composition of those two transformations—one  followed by the second. That's a very important   practical and theoretical problem.

Now there's  sort of a special case of this. The special case   is when the axes of the subsequent rotations  are pretty close together. So, for example,   suppose that one axis was like this—we 

had some rotation—and then another axis   just a little bit moved from the first one,  and then another rotation about that one,   and maybe a third axis also, again, close to  the original ones. So in this case, we have   perhaps three or more axes which are all more or  less in the same region of space.

In that case,   the various rotations that we have can be  viewed just on a part of the sphere's surface,   a little map of the region around the three  points, say the North Poles of the axes,   and then the rotations that we're doing here  are approximated by planar rotations in this   two-dimensional planar approximation of  the spherical surface near those points.

So what we're saying here is that if we want  to understand the composition of rotations on   a sphere or in three-dimensional space, probably  we should understand how planar rotations work   when we have centers of rotation that are  not necessarily the same point. So we have   a rotation about this point, and then maybe 

a rotation about this point, and a rotation   about this one. How do we effectively work with  products of rotations in the plane about different points? Okay, so there's a nice way of dealing  with that, and it's to recall that a   planar rotation can be specified by a pair of  reflections.

We saw this in our last video,   that a rotation corresponding to multiplication  by a complex number of unit quadrants can be   thought of in terms of a product of reflections  of two lines through the origin. So, for example,   here—this is all in the plane now—there's,  say, a center O, and here are two lines,   let's say L1 here and L2 here, and let's consider 

the reflections in those two lines. Let's call   them Sigma 1, the reflection in L1, and Sigma  2, the reflection in L2, and let's have a look   at the composition Sigma 1 followed by Sigma 2. In  other words, that sends a point X to Sigma 1 of X,   and then applies Sigma 2 to that.

What  happens to various points? Well, for example,   this point A—when we reflect it about L1, we're  going to get first this point right here, and   then when we reflect that about L2, we're going  to get this point over here. So the net effect   of this composition is to send A to A-prime over 

here. Suppose we choose some different points,   say U, this point B—when we reflect it in  L1, we're going to get some point over here,   for example, and then when we reflect that in  L2, we're going to get this point, point B-prime,   over here. And you can try some other points 

and convince yourself that effectively the   composition of these two reflections is a  rotation. It's a rotation about the point O,   essentially of twice this angle, where you can  think of the angle as not a measurement but   actually of that geometrical segment. Imagine 

putting that wedge together with another wedge   beside it, and you're getting the size of the  rotation formed by the composition of those two reflections. So that's a very interesting and important tool  in thinking about rotations—that we can break   them up as products of reflections, which are,  in some sense, more simple and more fundamental   kinds of transformations.

Let's start now then by  considering the product of two rotations in the   plane. We have a first rotation about the point  A1, called Row 1—that's rotation number one,   rotation around A1—and another rotation, Row  2, around the point A2, and we're interested in   describing what happens when we perform Row 1 and 

then Row 2. And let's suppose that we understand   Row 1 in terms of a pair of reflections—the  reflection in the line L1, and then the   reflection in the line L2. We can symbolize  that by this little arrow here.

So this little   arrow from the line L1 to the line L2 somehow  represents this rotation—the combined effect of   reflecting in L1 and then following by L2. That's  a rotation, effectively, of twice that angle about   A1. Similarly, this reflection Row 2—we're going 

to think about that as being the composition of   reflection in the line M1 and the line M2,  represented by this arrow, this red arrow. Alright, so how are we going to compose this  rotation followed by this rotation? Well,   the key is to remember that this pair of lines  that encodes the rotation is not unique.

If we   rotate both pairs of lines about A1, say we  take the initial one here and we rotate both   of them to get two lines like this, then  the rotation represented by this pair of   lines is the same as the rotation defined by  this pair of lines because this separation   between these two lines here is the same as the  separation between these two lines here.

So the   rotation of essentially twice that angle  is going to be the same as this rotation   of twice the angle there. Alright, that's  the key kind of flexibility that we have   in this kind of description of a rotation as a  product of reflections—that we're allowed to,   in fact, rotate the pair of reflections 

that we're using to define the rotation. So how are we going to use that to help us compose  these two? Well, what we're going to do is we're   going to rotate this pair, and we're going to  rotate this pair in a very special way. We're   going to rotate this pair so that the second 

line, L2, happens to go through not just A1 but   also A2. Alright, so we're going to take this pair  here, and we're going to rotate it around—so think   about this L2, we're going to move that around so  that it's now facing in that direction. There it   is there.

Then the L1 line will have moved  down here, so it will be in that position,   so we can call this L2-prime, and this one  L1-prime. These are the same lines here   after we've rotated them. And similarly with 

this representation of Row 2, we're going to   move that one around so that the first line, M1,  also has this property that it's this line right   here—that it joins this center to this center. So  we're going to take M1, and we're going to move   it around so that it's in this position, giving  us M1-prime, and then M2 correspondingly will   be in this position.

So this pair of lines will  have moved so that they're now in that position   there. So we've arranged it so that the new L2 and  the new M1, L2-prime, and M1-prime are the same. Why is that good for us? Well, because it's 

going to induce a cancellation of reflections,   for the simple reason that if you have a  reflection in a line and you do it twice,   then the effect cancels out—you reflect in a  line, and then you reflect in the same line—that's   the same as doing nothing. So, in other words,  algebraically, the rotation R1 followed by the   rotation R2 is going to be the composition of the 

reflection in L1-prime followed by the reflection   in L2-prime—that's what Row 1 is. And then Row  2 is the reflection in M1-prime followed by   reflection in M2-prime. So with this composition  business, we sort of read from right to left,   so the composition is a reflection first in this 

one, and then in this one, and then in this one,   and then in this one. And because these  two lines are the same, we've arranged   that it means these two reflections in here  cancel, and the whole thing just simplifies   to the reflection in L1-prime followed by the  reflection in M2-prime.

L1-prime was this one,   and M2-prime was this one. So the product of the  two reflections is over here—same picture, just   over here is the reflection in this line, followed  by the reflection in this line. These two lines   meet at a new point—let's call it A3, which is 

this point here in this diagram—and the original   points A1 and A2 now don't really play any more of  an important role. The two reflections in L1-prime   and M2-prime now generate a rotation about A3,  and we know exactly what that rotation is—it's   the rotation of twice the angle between these  two lines, L1-prime and M2-prime.

So this is a   calculus or way of computing with planar rotations  purely geometrically, in terms of pairs of lines   that encode the rotations via reflections. You  move the pairs of lines around so that the middle   two agree, and then they disappear, and then just  the outer two that determine the new rotation.

So we've described what happens when you  compose rotation Row 1 with rotation Row   2—the first one was a rotation around A1,  the second one was a rotation around A2,   and now we find that the product  is Row 3, the rotation around A3. Now it's proper for me to mention that 

sometimes something a little bit different can   happen. So these two lines, L1-prime and M2-prime,  that we obtained here through A1 and A2, in this   case, they meet at this point A3. It's also  possible for those two lines to be parallel.

Now   if those two lines had been parallel, then there  would be no point A3 where they meet, and in fact,   in that case, we would not get a rotation as the  product. The product of the reflection in a line   with the reflection in a parallel line is rather  a translation in their common normal direction.   Please make sure that you convince yourself of 

that—in fact, it's a translation by essentially   twice the separation between the two parallel  lines. In some sense, the center of rotation is at   infinity. Alright, so that can also happen, so we  don't necessarily get a rotation—it could be that   the product of two rotations is a translation in 

the planar setting. All this is a very geometrical   approach—it's a kind of ruler and compass  approach that if you actually had two rotations,   you could make diagrams and geometrically with  constructions find out where the center of the   product of two rotations is, and what the angle  or the amount of rotation about that new point   would be by computing these two new lines.

This  is essentially a Euclidean geometry construction   of how to multiply rotations in the plane. This  is going to motivate and give us some direction   for understanding the more complicated problem  of how to deal with three-dimensional rotations.

So let's apply what we've learned in the planar   case to try to understand rotations of  three-dimensional space—rotations of the sphere. So the key point is that a rotation of  three-dimensional space can also be described as   a product of two reflections about two planes  through the origin.

So let me illustrate that   here with this example sphere—I've taken an  equator, this one here now, and I've taken the   North and South Poles—there they are. Okay, and  we're considering a rotation around that axis,   and in particular, I'm going to consider a  rotation which is related to two planes that   go through this axis, and one of the planes is 

cutting the sphere along this orange great circle,   which goes through our North and South Poles,  and the other one cutting the sphere along this   blue great circle. And I want to consider what  happens if we reflect first in the orange plane,   followed by reflection in the blue plane. Let's 

have a look first at reflection in this orange   plane. So a reflection in a plane, like a mirror  image on the surface of the sphere—this fellow   here, who's sort of currently facing in that  direction—if we reflect him in the orange line,   we're going to get that fellow there—that's  his reflection.

Notice that the orientation   has been changed—if he was left-handed over  here, he would be right-handed on this side.   Reflecting in a mirror changes the orientation.  This point here, this little circle, for example,   would be reflected to this circle, and we  could compute the image of any point—of course,   points lying on the orange great circle  themselves are fixed under that reflection.

Alright, now let's follow that by reflection in  this second plane, the plane of the blue line   that also goes through the center of the sphere.  We're going to reflect in that blue line—then this   point over here, this little figure, will  get reflected to this one over here. Right,   so we're going from there to there—that's the 

reflection in the blue line. And similarly,   this circle here got first reflected here, and  then actually its image was here, not the one over   here—that's a mistake. So the reflection in the  blue one here is there, and then the net effect   is to send this little circle rotated to here.

So  overall, the combined effect of reflection in the   orange plane followed by reflection in the blue  plane is to rotate the sphere—rotate the sphere   by essentially an angle which is twice this angle  between these two lines. When I say angle here,   I probably should use a different word, but twice 

the separation contained by these two lines. Okay,   so a vector like this is not going to be  sent to there, but it's going to be sent   to there—the North and South Poles remain fixed.  So this is a way of representing a rotation of   the three-dimensional space as a product of  two reflections in planes.

And just as in the   two-dimensional situation, these two planes  are not really canonical—they can be moved,   they can themselves be rotated, so if we  rotated both of them around this axis,   we'd get a new set of pairs of planes which  would generate the same rotation that we started with. So in terms of our picture, here's 

a two-dimensional representation of   that. So here's our sphere, here's  a North Pole and a South Pole,   and there's an axis of rotation through those  two poles, and we've shown the equator of that   axis as well. So the plane of that equator is 

perpendicular to the axis that we're rotating around. Okay, and we're going to suppose that our  rotation that we're considering is generated by   two reflections about planes, and the first plane,  say, is given by this great circle here. Okay,   let's call that C1—that's a great circle on the 

sphere. It means that it's the place where the   sphere meets a plane that passes through the  center of the sphere—not just any old plane,   but a plane that goes right through the  middle of the sphere—that cuts out a   great circle. These are the biggest circles on 

the sphere. And we have a second such plane,   represented by this great circle here, going  behind the sphere and coming out like that,   and that's C2. So those are our two great circles.  So what we're doing is we're considering the   rotation that's the product of reflection Sigma 

1 in the first circle, C1, followed by Sigma 2,   the reflection in the circle, C2—that's  our rotation, we're going to call it Row. And let's make sure that we see what it's  doing. So, for example, this point A—when we   reflect it in the first great circle, we maybe get 

a point like this, and then when we reflect that   in the second great circle, we get a point like  that. So the cumulative effect is to send this   point A to this point A-prime. But again, there's  no requirement that the point is actually sent   along this curve—it could also, you can also think 

of it as having walked around the backside of   the sphere, or in fact traversed any old path to  get from here to here. The crucial thing is only   that we're starting here and ending here. What  about, say, the point B over here on the side of   the sphere facing us? If we reflect first in C1, 

we're going to get this point over here, and then   when we reflect that in C2, we're going to get a  point B-prime over there. So the cumulative effect   is to send this B to this B-prime, and that's  a rotation of the sphere, again by essentially   twice the separation of those two lines. So 

how are we going to describe geometrically that   rotation? Well, one way would be to describe it  by specifying the two great circles on the sphere,   but there's a little bit more efficient way  of doing that, and that's to use this equator,   which is another great circle perpendicular to  both of the great circles that we've just been considering. So the first circle, C1, has 

itself a perpendicular axis direction. So   if we think of the plane, C1, it has  a perpendicular axis which meets the   equator at some point—let's call it P1.  Okay, so this point, P1, is the point   on the equator which is like the North Pole or  South Pole if we think of this C1 as being a new equator.

Similarly, if we think of  C2 as being some kind of new equator,   then it has a North and South Pole, and these  are going to be, say, this point and this point   over here. That line would be perpendicular  to the plane that we used to cut out C2,   and those two points are going to 

also lie on the original equator,   which is the equator corresponding to the North  and South Poles where the two planes C1 and C2 meet. So how are we going to represent  this combination of reflections? We're   going to do something different than  we did in the planar case.

We're not   going to use the two lines or the two great  circles passing through our North Pole—we're   going to rather use these two points,  P1 and P2, on the equator of the axis of rotation. So there's P1, there's P2. You might 

say, well, which P1 are we going to use? There's   actually two of them. And which P2 are we going  to use? There's also two of them. That's right, so   we're going to choose one of these P1s—it doesn't  really matter—and we're going to choose the P2   which is closest to the P1.

One of them will be  typically a little bit closer on the equator than   the other, and so the shorter arc, we're going to  use that as the arc to represent this rotation,   and we're going to actually draw a little  arc in red on this equator, joining P1 to P2,   and that little arc will contain the information  of this rotation.

In fact, what the rotation   ends up being, it ends up being a rotation by  essentially twice that arc, with this as its equator. Alright, so there's quite a lot in this  diagram. It's much easier to visualize if   you get yourself a ball—it could be 

a soccer ball, basketball, a globe,   beach ball—get some kind of ball and try to  essentially recreate this picture and make   sure you understand where all the various  points are. Okay, it's a bit complicated,   but a very rich picture, and it's sort of an  interesting and non-intuitive idea that we're   going to represent this rotation about this axis 

here by these two points. So this little vector,   joining these two points on the equator of that  rotation, and we have some flexibility about doing   that because we can rotate these two points along  the equator—we can put them wherever we want along   the equator as long as they're separated by the  same amount, they're going to represent the same rotation.

So to come back to our original picture here,  I guess we had this story like this where we   were rotating about the North and South Pole  here—we're rotating that to send this vector   over to some place like that, to go from here  to over there. And so we're thinking of that as   being the composition of the reflection in this 

plane followed by the reflection in this plane.   So how would we represent that? Well, we could  take our first plane, which is the orange one,   and find a normal to it, so the North Pole  essentially of that great circle—maybe that   one—I'll make it big—that'd be the first  one there. Alright, and then we're going   to look at the second plane, which is the 

blue plane—it also has a North Pole and a   South Pole—maybe that one there, I'll choose.  Okay, that's the North Pole for the blue line,   and that's the North Pole for the original  orange line, and then the representation of   the rotation that we're considering is this point  and this point, followed by being connected with a vector. Okay, so that vector joining that point 

to that point on this equator represents the   following rotation—it represents the rotation by  essentially twice this amount, so something like that. Right, and it doesn't have to be this  vector—we could move this vector over, so I   could move it over so it's, say, between here and  roughly an equal amount, say over there.

Right,   so this vector on the sphere would represent  the same rotation as this vector—they would   both represent the rotation by essentially  twice that separation, with this as the axis. That's a geometrical way of representing  rotations of three-dimensional space,   by spherical vectors on the surface of the 

sphere—vectors which lie on great circles,   essentially on the equatorial great  circles of the rotations we're considering. So now we can use these rotation vectors for  rotations to actually describe the composition   or the algebraic structure of rotations of  three-dimensional space.

And we can do so   in a very geometric and direct way, which is  completely analogous to what we were doing in   the planar situation, even though the object that  we are using are a little bit different. Alright,   so here's how it works: we have two rotations of  our sphere, or space, and they're represented by   these two rotation vectors, Row 1 and Row 

2. Alright, so Row 1 is the vector which is   lying on this equatorial great circle—that's  an equatorial circle for the rotation Row 1,   which actually rotates by twice that separation.  So it would send this point to that point there—a   rotation with this as the equatorial direction,  and the axis would be somewhere like this.

And   then we have a second rotation represented by  this little vector, which is also lying on a great   circle between this point and this point, and  that represents a rotation with an axis in that   direction somewhere, essentially through twice the  separation here. So it would send this point to   a point around the other side.

Alright, so these  are the objects that represent the two rotations,   and to combine these two rotations, we're  going to rotate both pairs so that the   final point of Row 1, this one here, and the  initial point of Row 2, this one here, match up. Alright, so we're able to do that because 

this Row 1 vector represents the same rotation   as the one that we would get by moving that,  sliding it down so it would be in that position   there. Alright, so I'm going to slide this vector  down until it's in that position there—it still   represents exactly the same rotation as before,  it's just that it's now in a different position   along its equator.

And I'm going to slide this  vector, Row 2, along its equator so that its   initial point is at the same place where the final  point of Row 1 is, where the two equators meet. And why does this help us? Well, it's because the  rotation Row 1 that we're talking about can be   thought of in this way: the rotation is a product 

of two reflections, and the reflections are the   reflection in the plane which is perpendicular  to this point. So this point here has an axis   coming out from the center of the sphere, and  there's a perpendicular plane to that direction   from the center of the sphere, which determines  a great circle—that's the first reflection that's   involved in Row 1.

And the second reflection  is the one in the plane perpendicular to this   point. So the combination of Row 1 and Row 2  is a reflection in the plane perpendicular to   this point, followed by a reflection in the  plane perpendicular to this point, and then   Row 2—reflection in the plane perpendicular to 

this point, followed by reflection in the plane   perpendicular to this point. And just as in the  planar case, those two intermediate reflections   cancel because they're the same reflections—we've  cooked it up so that they are the same. So   the product reduces to just a product of two 

reflections in planes perpendicular to points,   namely to this point and to this point. So  the composition of these two rotations ends up   being the rotation determined by reflecting  in this point—actually, reflecting in the   plane perpendicular to this point, followed by  reflecting in the plane perpendicular to that   point.

And that's given by the vector—the vector  on the great circle joining these two points. So that's the algebra—you take a spherical  vector like this, you move it down,   and you move this one across so that they're  lined up just like an addition of vectors in   the plane.

And we're kind of adding these  vectors just as we do vectors in the plane,   except it's all happening on the surface of the  sphere. So we get this vector plus this vector   equals this vector here, which I haven't shown  here, but here it is.

And that corresponds to   this rotation times this—this rotation is equal  to this rotation. So we see that geometrically,   rotations can be thought of in a very, very simple  way if we are happy to use these spherical vectors   lying on the sphere, and if we're happy to  agree that rotating a spherical vector along   its own direction doesn't essentially change 

it. Alright, that's an interesting kind of   algebra. So we end up expressing the product  of rotation Row 1, Row 2 as the rotation Row And notice that special kind of thing that  happened in the plane where we ended up   getting two lines which don't meet—it's 

not going to happen here because on the   sphere, any two great circles are  guaranteed to have an intersection  point. Get a sphere, play around,  convince yourself this works. So great, we've basically given a demonstration 

of this important theorem going back to Euler   in 1776, which is that the product of two  rotations is a rotation. In other words,   if you take your sphere and you take any axis  and you perform one rotation, and then you take   some other axis at random and you perform another  rotation, then that composite transformation—going   from some initial position to some other initial 

position—is actually a rotation. Somewhere,   there's an axis so that you can essentially  get back to where you started just by rotating   around that axis. It's not entirely obvious that  that's the case, and Euler realized that it had   to be proven, and he proved it.

And so basically,  we've given a geometrical demonstration of that   fact using this idea of spherical addition  of vectors on the surface of the sphere. Okay, so that's all very geometrical with pictures  and models and so on, which is great.

But there's   also a need for having an analytic approach where  we use coordinates. And this is a more familiar   approach, I suppose, because people are very  familiar with linear algebra. If you've done any   mathematics at the university, you've probably 

taken a linear algebra course. If you haven't,   and you'd like to learn some linear algebra,  may I recommend my series Wild Lin Alg, which   lays out linear algebra in a more geometrical way  than most courses, and it's ongoing. So in fact,   the second half of this course, which is going to 

be starting soonish, will be talking a lot about   the kinds of things that we're talking about today  and will give a lot more detail and derivations.   So if you're interested in this kind of thing, you  should check out this series—especially the second   half. Of course, you should have already watched  the first half before you watch the second half.

Okay, so we're going to introduce some  basic things from three-dimensional space,   in particular, first coordinates. So  we have an X direction, a Y direction,   and a Z direction—these are three mutually  perpendicular lines.

You can think of the Y   direction being in the plane of the board, the  Z direction is being in the plane of the board,   and the X direction coming straight out—it's  just that you're sort of over there somewhere,   and you're looking at it in that direction,  so that the X direction appears something   like this to you. Then we use these coordinates 

to specify the position of any point in space,   say the point P with coordinates A, B,  C—is that point whose X coordinate is A,   whose Y coordinate is B, whose Z coordinate is  C. And it's best thought of in terms of a box   whose sides are parallel to these axes. And here's 

the box here, kind of going A in the X direction,   B in the Y direction, C in the Z  direction—that's the point A, B, C. Alright, so what we want to do is we want  to apply some metrical considerations to   this three-dimensional space. So we want to take 

planar notions that are metrical and extend them   to three-dimensional space. So the fundamental  planar notion is the notion of quadrants. Okay,   so if you have, say, consider the vector or the  segment from zero to this point R here, which   just in the plane has coordinate A and B, then 

its quadrants, the quadrants between O and R, is   A squared plus B squared. Notice that we're using  this quadrant and not its square root—the square   root would usually be called the distance or the  length, and we want to avoid that because taking   a square root is a transcendental operation. It's 

usually called algebraic, but it's not algebraic   at all—it's actually a transcendental infinite  process that never stops. You can't actually   complete a calculation of a square root—that's  just the reality of things. We want to avoid   that—that's too complicated.

Quadrants, A  squared plus B squared—very nice and simple. How do we extend that to three dimensions?  Well, once we have the quadrants here,   there's another sort of right triangle going  up C.

So we have a quadrant of A squared plus   B squared here, a quadrant of C squared here,  because the coordinate is C, so the quadrant   is C squared. And so Pythagoras's Theorem applied  to this triangle here tells us that the quadrants   between O and P should be the quadrants between O  and R, which is A squared plus B squared, plus the   quadrants between R and P, which is C squared, for 

a combination of A squared plus B squared plus C squared. So we are extending the definition  of quadrants to three dimensions. We can   define the quadrants of a vector A, B,  C to be A squared plus B squared plus C squared.

So motivated by Pythagoras's Theorem, we  can define this to be the quadrants of a vector,   and that's the fundamental metrical  notion in three-dimensional space. The next important notion is perpendicularity.  How do we tell or how do we define what it means   for two directions to be perpendicular? Alright, 

we'll work in the language of vectors. So suppose   we have one vector, V1, and so the vector is  now actually something with a start and an end,   so it's a directed line segment. I denote  it with round brackets because it's subtly   different from the point with square 

brackets. So when you see round brackets,   I'm usually talking about a vector. So we have  vector V1, X1, Y1, Z1, and another vector, V2,   X2, Y2, Z2, and we want to ask, well, when  are these perpendicular vectors like these ones? Well, if we look at this triangle here in 

vector form, this one's V1, and this one's V2,   then that vector will be V2 minus V1. And if  we have perpendicularity here, then we expect   that Pythagoras's Theorem should apply to this  triangle, which means that the quadrants of V1   plus the quadrants of V2 ought to be the quadrants  of V1 minus V2, or V2 minus V1—it's the same.   So what does this equation relating the three 

quadrants amount to in the coordinates? Well,   quadrants of V1, we've defined it to be  this—quadrants of V2, we've defined it to   be this—and the quadrants of the difference?  Well, we have to take the vector, which is the   difference of the coordinates, and we have to take  the sum of the squares of those coefficients. So   this equation represents perpendicularity in space 

if we want Pythagoras's Theorem to be true. And   we want Pythagoras's Theorem to be true in three  dimensions, just as it is in two dimensions. So   fortunately, this equation appears rather long  here, but there's a lot of cancellation.

Once   you expand the right-hand side, because the X1  squared will cancel with that, and the X2 squared   will cancel with that, the only thing that will  be left on this side after you divide by two and   maybe take out a minus sign is the expression  X1 times X2 plus Y1 times Y2 plus Z1 times Z2,   and that's got to be equal to zero because that's  all that's going to be left on the left-hand side.

So I'm showing you now that if you want  Pythagoras's Theorem to be true, then you want   this relation to hold between the coefficients  of V1 and the coefficients of V2. This is a   very important expression in the coefficients  of V1 and V2 for exactly this reason.

Okay,   this is the reason why this is important. Now,  of course, in linear algebra, they just sometimes   introduce this and say, well, here's the dot  product—we're just going to define the dot product   without telling you why you want to know about  the dot product.

Okay, I'm telling you why this   is important—it's important because it's going to  guarantee that Pythagoras works in this situation. So here's the definition: V1 dot V2 is X1 times  X2 plus Y1 times Y2 plus Z1 times Z2. And this is   a number.

So given two vectors, we get a number,  and that number has different names—sometimes it's   called a dot product, sometimes it's called  an inner product, sometimes it's referred to   as a bilinear form—all valid names for it. Very  important object, and basically, we've established   that two vectors are perpendicular precisely when 

V1 dot V2 equals zero, where perpendicularity is   essentially Pythagoras's Theorem. Of course, if  you don't want to introduce Pythagoras, you could   say, well, we're just going to introduce this and  then we'll define this to be perpendicularity,   and then you could prove Pythagoras's  Theorem—that's also a reasonable approach,   maybe actually cleaner algebraically.

But  it's important to realize the motivation   for this is because of this computation  linking Pythagoras and the bilinear form. Now, there's another important construction that  we can do with vectors in three-dimensional space   that we can't really do with vectors 

in two-dimensional space or, in fact,   very easily with vectors in any other dimensional  space—something special about three dimensions,   the world in which we live in—and that's the cross  product of two vectors. Here's the definition:   so we again have vector V1 and V2 with  coordinates just as before, and we're going   to define this particular combination, and this 

time I am just going to pull it out of the blue,   but I'm going to explain why in a second. Alright,  so it's a new vector. So the cross product of two   vectors is a vector, not a number like the dot  product, a vector, and it has three components,   and they are: the first component is Y1 times 

Z2 minus Y2 times Z1, and then the next one   is sort of obtained cyclically from this  by replacing Y's with Z's, Z's with X's,   and X's with Y's. Alright, so we get Z1 times  X2 minus Z2 times X1, and then replace Z's with   X's—X1 times Y2 minus X2 times Y1. Alright, so 

for those of you who know some determinants,   you can write this in terms of a determinant, and  it's another vector, and it geometrically has the   following properties: that if, say, there's the  vector V1 and there's the vector V2, that V1   cross V2 is a vector perpendicular to both V1 and  V2. And furthermore, its quadrants are intimately   connected with the area of the parallelogram 

formed by V1 and V2, so the bigger that area is,   the wider V1 and V2 are apart, in some sense,  then the bigger the cross product is going to be. So I'm going to just state a  few properties linking the dot   product and the cross product—these are pretty  familiar formulas in a linear algebra course,   and in the Wild Lin Alg course, we are going 

to talk more extensively about these and   other properties of the dot product and cross  product. So the first is that V1 cross V2 is   really perpendicular to both V1 and V2, and it's  going to be an exercise for you to check that.   So what you have to check is that when you take  the dot product between this one and this one,   you get zero, and when you take the dot 

product between this one and this one,   you get zero. That's what perpendicularity  means in the linear algebra context. The   second property to check is that if we change  the order of the vectors, then the cross product   changes by a sign.

The next thing is that  it's bilinear, so that V cross U plus W,   for two different vectors U and W, is going to  be V cross U plus V cross W. And then there are   some more fancy, interesting properties of the  cross product—the first is Jacobi's identity,   which is that V cross U cross W, now that's 

going to be another vector—U cross W is a vector,   and you're allowed to take the cross product with  a vector and that one. So that combination plus   U cross W cross V plus W cross V cross U—they  all amount to zero. Maybe I should say the zero vector.

Another is that the quadrants of V cross  U, so you take the quadrants of this vector,   you're getting the quadrants of V times the  quadrants of U minus the dot product V dot   U squared. It's a very important formula,  something that's called Euler's identity or   closely associated with Euler's identity.

And then  there's a generalization of this, which is that U   cross V dot W cross Z, when we have four vectors,  can be written as U dot W times V dot Z minus   U dot Z times V dot W. You might like to think  why this is a generalization of that.

Alright,   so these are exercises for you to do  to practice your algebraic skills. And many of you will have learned about dot  products and cross products and connected   them with angles, formulas involving sines and  cosines—may I humbly suggest that you not think   in that direction.

Okay, that is not really  going to help your ultimate understanding of   this subject. I'm sorry to say, but I must say  that. Okay, the cosine and sine business is a   way of introducing transcendental aspects where 

they don't properly belong. Okay, so this is a   purely algebraic story—that's the way God meant  it to be. Okay, it's purely algebraic—high school   algebra just rolls out if you do it in the right  way.

Okay, of course, there's a lot more to be   said about the dot product and cross product, but  this is enough of an orientation to prepare us for   our understanding and appreciating Hamilton's  discovery of quaternions. And there's a lot of   interesting twists to this story—in Hamilton's  time, this notion of dot product and cross   product was not around.

It's very familiar to  modern students, especially if you've taken some   linear algebra at the university level, or some  physics because these notions are very important   in physics. But in Hamilton's day, they were not  around, and his discovery of quaternions, in fact,   ultimately motivated people to think about the dot 

product and cross product, which are intimately   connected with the algebra of quaternions and  ended up replacing quaternions in people's   minds—but not before a long battle ensued between  the proponents of quaternions and the advocates   of the vector calculus that I'm describing  here. So the history is quite interesting.

So in our next lecture, we're going to talk about  Hamilton's discovery of quaternions—we're going   up to four dimensions now. Two dimensions  first, now three dimensions, next time four   dimensions—this beautiful algebraic structure  that sheds a lot of light on what's happening   here in three dimensions.

I hope you join me  for that. I'm Norman Wildberger, thanks for [Music] listening.