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Classical to Quantum | For rational 3D rotations - start with reflections | Wild Egg Maths
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We begin to create a novel 3D rotation theory-- entirely rationally, without any "irrational numbers" like "pi" or "e", or "transcendental functions" like "sin" and "arccos"; relying instead on complete and fully correct computations that can be done on our machines with 100% precision. A starting point is to review some basic linear algebra involving a symmetric bilinear form and the associated quadratic form (quadrance!), along with notions of projection of a vector onto another vector or an orthogonal hyperplane. In three dimensions reflections through planes through the origin are particularly useful building blocks for more general orthogonal transformations. The rational parametrization of the unit sphere of course also will play an important role. Note: On slide 6, I mistakenly talk about "rotations" when I am actually referring to "reflections". The "sigma's" will always be reflections for us. This topic is very important for a variety of practical areas, including not just physics, but also chemistry and crystallography, computer graphics and CGI animations, robotics, aerospace engineering, computer vision and signal processing. It is also an important stepping point to understanding relativistic geometries and transformations more correctly. This video is in the Classical to Quantum series, available here for Members of the Wild Egg Maths channel.
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Show transcript
0:01
thanks for joining me on this adventure we're going to be talking today about rotations in three
0:08
dimensions now which is a big step up from the two-dimensional Theory we've been talking about already and this is
0:16
um very important of course for physics but also for a lot of adjacent areas uh
0:22
chemistry uh computer Graphics video game design robotics even Optics and
0:30
architecture and a lot of industrial applications anything where we're moving around and we are utilizing rotations
0:38
and we want to have a nice algebra of rotations so we're going to Aspire to
0:43
laying out such a theory without any approximations so no irrational numbers
0:49
no cosiness or signs no functions that cannot be evaluated completely so we
0:55
want the entire Theory to be on the page fully 100% And we want to be a support
1:01
the theory and uh the developments in a complete way with explicit examples so
1:08
the setup is really basic linear algebra we're thinking about working in a
1:13
three-dimensional Vector space the default is for us the rational numbers as the underlying number system and the
1:20
metrical structure of this three-dimensional space is encoded by a symmetric bilinear form sometimes called
1:26
a DOT product or inner product and here it is we have two vectors V1 and V2 the
1:32
dot product if the coordinates are X1 y1 Z1 and X2 Y2 Z2 then the dot product is
1:38
the number x1x2 + y1 Y2 plus Z1 Z2 so this is a bilinear form it's linear in
1:45
each of the sets of variables individually and Associated to that there's also a quadratic form where we
1:52
look at just V Dov so quadrants of V is x2 + y2 + z s we're going to use this
1:59
term quad quadrants from rational trigonometry and in fact rational trigonometry is going to be playing a
2:04
very important role as we proceed uh very Central importance is
2:10
the unit sphere S2 just those vectors whose quadrants is equal to one and it's
2:18
important to uh just note that there's something special about this symmetric bilinear form and that it's positive in
2:24
the following sense that the quadrants of any Vector is always greater than or equal to zero and we have equality the
2:31
quadrants is zero only in one case when the vector v is itself the zero Vector
2:38
all right so this is the setup and of course it's a mathematical manifestation
2:44
of our three-dimensional space and here is kind of a model of the the unit sphere and what we want is to ultimately
2:53
understand rotations and of course geometrically we all sort of know what a rotation is you choose two antip points
3:00
and imagine an axis through the sphere going through the center of the sphere and then we're looking at
3:06
Transformations which are fixing those two antialbuminuric
3:29
up with a cumulative thing which is a third rotation so the relationship between those rotations is not at all
3:36
obvious but a very important sort of practical uh thing that we want to get our heads around so we're going to try
3:42
to understand this entire Theory purely in an algebraic or rational way okay and
3:50
for us the key will be to think about Reflections
3:55
first so a pleasant starting point though is to just look at the unit sphere first and obtain a rational
4:03
parameterization of it now if we look at the ordinary sphere like the Earth okay
4:08
we all know that uh the standard way of parameterizing or labeling positions on
4:14
here that we commonly use um I guess we should Orient it like this uh is in
4:19
terms of longitude and latitude and those are given in terms of sort of angular measurements uh from some
4:26
specified Direction so we have uh longitude going around in this direction you know from 0 to 360 and then we have
4:32
a kind of a latitude going from 0 to 90 and 0 to 90 down in this direction so
4:38
angles are sort of the familiar way of parameterizing or mapping or
4:51
coordinatization parameterization has the advantage that we can work with it completely uh with
4:58
full arithmetic so two parameters T and R these are just
5:03
rational numbers and we Define e of t r to be this triple okay 1 - t2 - r 2
5:10
over 1 + t2 + r 2 comma 2 T over 1 + t2 + r 2 comma 2 R over 1 + t2 + r 2
5:21
and the picture that you should have in mind okay here's our three-dimensional space the X and the Y coordinates I'm
5:27
going to put them in the plane of the board so they're the usual XY
5:33
coordinates that we're used to and our third axis are sort of coming out so we have a right handed coordinate system x
5:41
y and Zed coming out but we'll picture it like with the Zed axis coming in this
5:46
direction here this is the positive Zed Direction This is showing us the unit
5:51
sphere coordinates 1 minus one 1 - one and 1 - one on the three
5:57
axes so what's the meaning of this well the meaning is that we start with this
6:02
point here on the xaxis at coordinate minus1 very much the same thing that we do in parameterizing the circle and we
6:10
look at this plane the x equals z plane which is the plane of the Y and Z axis
6:18
and we think of TR as being a point on that plane like say that's the point t
6:27
uh t with y and Zed coordinates t r so in terms of the three coordinates it'll be zero in the x and t in the Y and R in
6:36
The Zed okay so that's a point in this plane and then what we do is we join this point to that point in the plane we
6:42
get a line and we look at the other point of intersection of that line with
6:48
the sphere and that's what we call E of TR and it's a calculation to show that
6:54
that point is exactly this thing here now what makes this a point on the
6:59
sphere is this algebraic identity that 1 - t2 - r 2 2 + 2 t 2 + 2 r 2 2 is
7:10
1 + t2 + r 2 2 and you can sort of see this as true
7:15
without even expanding it too much because if we expand say this side here
7:21
we get the squares of those entries here which we're also picking up when we Square this thing here but we're also
7:28
getting twice the various products so twice this product twice this product and twice this product and that equals
7:33
what we're having on the other side because say the the 2 t2 that appearing here um is happening here because when
7:41
we expand this we get minus 2 t2 but then we picking up a 4 t2 there for a
7:47
total of 2 t2 and the same thing happens with the 2 R 2 and the two t2 R
7:52
2 are appearing just as it is so this is an identity that's responsible for the fact that this thing really is a point
7:58
on the unit sphere now just as in the planer case
8:05
there's a slight hiccup with what happens to this particular point exactly because it's playing a special role in
8:12
this story The Point minus 1 0 0 so a natural
8:18
question would be can we do something like we did in the planer case sort of
8:26
projectivization uh which includes this point here as well so that turns out to be a very interesting question I'm going
8:33
to let you think about that we will come back to that important question in fact it's going to um thinking carefully
8:39
about that question is going to open up uh some new directions for projective geometry for us okay but this is a good
8:46
starting point so that if we want some unit vectors on the unit sphere we just
8:51
pick arbitrary rational numbers T and R and plug them into here and we going to
8:56
get unit vectors on the unit sphere on the algebraic side to capture
9:04
the rotations that we're interested in or perhaps the reflections we look at the orthogonal group now
9:11
03 those are the 3X3 matrices that satisfy the same orthogonal relationship
9:18
that Q * Q transpose is the three-dimensional identity
9:23
Matrix this orthogonal group then ends up splitting as a union of s O3 those
9:30
ones with determinant + one and let's say R3 those matrices with determinant
9:37
minus one or one bar as we sometimes prefer to say and just as in the planer
9:42
case and in the previous video we discussed the relationship between these
9:47
two things this is a subgroup and this is its coet just as what happens in the two-dimensional case in fact it happens
9:54
for an arbitrary Dimension this coet can be obtained from the subgroup by
10:00
multiplying uniformly by some given element in the coet so we can pick any
10:09
element say m and uh and then multiply elements of s SO3 by this m to get
10:15
elements of R3 so all would we need is a matrix with determinant minus one so
10:24
here is an example so this would be a candidate this would be somewhat analogous to what we did in in the
10:29
two-dimensional case the diagonal matrix with entri is 1 1 minus one but actually
10:36
of course we could also put the minus one there or or perhaps there so there's some Choice there and after we think
10:42
about it for a little while we realize that there's actually perhaps another Choice which is maybe more symmetrical
10:50
okay which is to not use this one but to use this one let's make each of these
10:56
diagonal entries minus one or one bar then the determinant will be still just
11:03
the product of those three minus 1es it will still be minus one this is um a simpler Matrix because
11:09
it's just the negative of the identity Matrix and multiplying by that Matrix is
11:15
very simple it just question of taking the negation of whatever Matrix we have here so when we
11:21
think about it this way we realize that this ro3 just consists of the negatives
11:27
of all the elements in s SO3 okay so that's something that happens in
11:32
this three-dimensional situation which wasn't happening in the two-dimensional
11:39
situation so why are these orthogonal matrices important well it's because they preserve the metrical structure so
11:47
if we look at the dotproduct that we've already defined and we multiply one of
11:53
these vectors by a an orthogonal Matrix on the right so we sent V to V * Q let's
12:01
call that V Prime then if we have two vectors then their images under this
12:07
transformation will have the same dot product as the original ones okay so
12:13
this is in metrical geomety this makes this action by q a rigid
12:20
motion and let's just remind ourselves that uh to connect the dot product with
12:26
this sort of Matrix theor that we've got here another way of thinking about the
12:31
dot product is just as this product between the vector V1 and the transpose
12:37
of the vector V2 so here is Vector V1 as a row Vector here is V2 as a column
12:45
vector and if we multiply those two things the way we multiply a row vector by a column Vector we get X1 * X2 + y1 *
12:53
Y2 plus Z 1 * Z2 so that's exactly the dot product that's really great that means that we can when we're working
13:00
algebraically with the dot product we can replace this dot product with just this matrix
13:05
product with a suitable transpose in the second Factor all right so what do these
13:12
orthogonal matrices look like well an important kind of orthogonal Matrix is a
13:18
rotation so we have to Define what rotation is and remember that I said when we were talking about this thing
13:23
here I said that basically a rotation is defined by having an axis which is fixed
13:28
and then then it moves the rest of the sphere around but keeping the this point and its
13:59
back again okay now we come to a a very important classical result which we're
14:05
going to state in the language of linear algebra but it's effectively a result of
14:10
Oiler and that is that in fact every element of SO3 is a
14:15
rotation every 3x3 Matrix which is orthogonal and has determinant one in
14:22
fact is a rotation in other words it has a nonzero fixed vector
14:29
that's not entirely obvious okay and one of our aims as we go along is to try to
14:34
understand why that actually is true I just want to mention at this point and I also want to mention this
14:40
sort of obvious coroller that the product of two rotations is then necessarily also rotation because we
14:48
know this is a subgroup of O3 and if you have two elements in here the product is
14:53
also in here so if everything in here is a rotation then the product of two rotations is necessary neily another
15:00
rotation and that's not obvious at all right I mean you know you've got one axis here and you
15:07
rotate and then you choose another axis and you rotate and then you compare this
15:12
final position to the original position what we're saying is that you can get from the original position whatever it
15:18
was to this final position by a single Rotation by finding somewhere another
15:24
fixed point and just rotating around that third axis
15:29
that's going to be an important sort of practical question how do you get that third axis if you know the two original
15:39
rotations and so that's the question that we're asking here is how do you describe rotations geometrically and
15:46
what's their algebra you see in the classical approach that most physicists will be familiar with a rotation is
15:52
determined by an angle right a rotation just described but you have a an axis
15:58
like this and then you're going to say okay we're rotating by 37.2 degrees or
16:04
maybe you know 089 radians if you're working in the radian measure and that's how rotations are described in terms of
16:13
angles but angles unfortunately are transcendental quantities that we can't in general
16:20
calculate so we don't want this Theory okay it's very useful practically and
16:26
I'm not diminishing its role both both historically and practically but in terms of a
16:33
theoretical mathematical framework that's going to be um you know sort of the right the right approach for a lot
16:41
of physics a lot of fundamental physics we would much prefer an alternate way of
16:47
thinking about a rotation where we can describe the amount of rotation in some
16:54
kind of rational way well happily we have trigonometry to
17:00
help us okay so um we are going to be able to to solve this problem but it's
17:06
um it's going to mean that we're going to step outside the familiar Zone that
17:12
you're almost all probably used to we're going to have to rethink what's really going on and restate things in a
17:19
rational form avoiding angles that's a big
17:25
challenge okay so how are we going to describe rot s and their
17:30
algebra well one thing that I could do is at this stage invoke the K transform
17:37
which I've already told you about it's a it's a way of sort of getting at orthogonal matrices from skew symmetric
17:43
matrices but there's an element of sort of pulling a rabbit out of a hat with that thing um and it's not you know
17:50
that's not a result that you could easily discover on your own if unless you have quite a lot of experience in
17:55
this in this area what i' prefer prer is to proceed in a more perhaps natural or
18:02
simple-minded way and have the KY transform sort of emerge as a
18:07
consequence of simpler kind of steps so that's what I'm going to be explaining to you and the way to do that is to take
18:16
a a page from what we did in two dimensions in two Dimensions we saw that we had this exact correspondence between
18:23
two-dimensional rotations and two-dimensional Reflections there was in fact an exact corresp respondence
18:30
there that exact correspondence breaks down in three dimensions as as it turns
18:35
out in fact I can explain to you why that might be useful to to mention okay
18:41
so if we think about rotations right and we think about you know how big is the group of rotations
18:49
well a rotation is given by essentially two things an axis Direction maybe that's the axis okay and some kind of
18:58
measure of rotation we're not going to use angles we're going to use something else okay whatever it is but there's a
19:04
kind of a one parameter family of rotations given that you have an axis
19:10
and how many degrees of freedom is there in choosing an axis well an axis can
19:16
more or less be chosen by choosing any point on the sphere and then it also its anpal point so you sort of have a two
19:23
dimensional family or choice of possible axes and a one-dimensional choice of
19:29
rotation amounts however you're going to measure rotation amounts that amounts to
19:35
a three-dimensional sort of family of rotations how about Reflections
19:42
well um a reflection is given by in this case like a plane you you choose a plane
19:49
okay and then you just reflect in that plane like a mirror and how many degrees of freedom
19:56
are there in that well a plane is determined because plane is going through the origin it's got to be sort of cutting the sphere symmetrically it's
20:03
determined by its normal Vector so there's a two parameter family of normal
20:09
possible vectors to planes and therefore a two-dimensional family
20:14
of Reflections so this is different from what happens in two Dimensions there's
20:20
actually quite a lot of differences between the two dimensions and the three dimensions one of the reasons why this
20:25
this whole subject is is very delicate and has has really been properly understood yet okay so the rotations are
20:33
some kind of three-dimensional object and the reflections are some kind of two-dimensional object so we don't
20:38
expect a correspondence but it will turn out that the reflections are simpler to get a hold of right we can get our hands
20:46
on the Reflections in a simpler way and then it'll turn out that we can get the rotations just by composing two
20:53
Reflections much the same as what we did in the two-dimensional case
20:59
so this is a very important order of business is to try to understand Reflections so for that we have to go
21:05
back to sort of first year linear algebra and do just a little bit of fiddling around with projections and
21:11
Reflections okay so we're working in this Vector space actually doesn't
21:17
really depend on the vector space being three dimensions this is just very general and it's useful to make that
21:24
that statement because uh this is going to be a very general um
21:29
construction that I'm creating here which has all kinds of uh implications
21:35
in higher Dimensions as well Okay so we've got a vector space we have a nonzero vector v here's the vector v
21:42
okay imagine that's the origin of our Vector space and Associated to this Vector
21:48
space we have this this um hyper plane which consists of those vectors W
21:56
which are perpendicular to V perpendicular means that the dot product between W and V is zero so in the
22:02
threedimensional case which is the picture that you're seeing here the hyperplane is a hyper plane it's a
22:09
two-dimensional plane sort of orthogonal to the to the vector v okay and now
22:15
let's let U be an arbitrary Vector in V there's U okay so then what we're
22:22
interested in is decomposing this Vector U into two components one component in
22:27
the v Direction and the other component in the perpendicular Direction in other words in the H subv Direction we'll write it
22:35
like this so we're going to write U as Alpha time V so this is a vector in the
22:41
direction of V but it's multiplied by some Factor some number in our field F
22:47
stands for our field the default for us is the rational numbers but this argument does
22:55
generalize okay so there's a component in this direction Alpha * Z and then there's something else W and we want
23:02
this W to be in this perpendicular Direction in other words we want w. V
23:08
equals z so the way we can get at that is just then by taking this equation and
23:13
dotting with v so we get u. V on the left hand side equals Alpha v.v plus w.v
23:21
because the product is or the bilinear form is linear but we're assuming that this W do
23:28
V is zero that W and V are perpendicular so that term drops out and
23:34
we just have Alpha time V Dov okay but that means we can solve for
23:41
alpha alpha is equal to U Dov which is a number divided by V Dov which is a
23:47
number and crucially V Dov is nonzero because we've chosen our vector v to be
23:53
a nonzero Vector remember I told you that if a vector is nonzero then its
23:58
quadrants this just the quadrants is also non zero so we don't have to worry about the denominator being zero here or
24:05
not okay so we've got Alpha and now we can return to this equation and maybe rewrite it in terms of w w = u minus u.
24:13
V over V do V * V so this is an expression for the perpendicular
24:21
component to V in U all right so we have this picture here there's Alpha time V there's W and these
24:29
two vectors are the components of U great and now we can use this diagram
24:35
to Define two related sort of reflected vectors one of them we're going to call
24:42
Sigma subv of U and the other one we're going to call row sub V of U so the
24:48
sigma sub V of U is going to be the reflection of vector u in this hyper
24:54
plane okay there's the hyper plane there's the vector U Its Reflection is going to be the vector
25:00
here the mirror image and what does that mean it means that we're taking this Vector here we're looking at its
25:07
component in the v Direction and we're subtracting that not once but twice so we're taking U and we're
25:14
subtracting twice this component is Alpha time V what was alpha alpha was u. V over v.v
25:23
so this is just u - 2 * u. V over V do V
25:29
* V this is a crucially important formula
25:34
okay it's the formula for the reflection of the vector u in the hyperplane
25:41
determined by V okay now there's another kind of
25:48
reflection that we also could do we could take the vector U and subtract the
25:54
other component so we here we subtracted um Alpha V twice we can also subtract W
26:01
twice that would mean we're going in this direction so we subtract W once and twice and we get another Vector called R
26:10
subv of U which is like a reflection in the vector v
26:16
itself well so that's U minus 2 *
26:23
W there was W so it's uus 2 * u - u. V over v.v *
26:32
V and now a little bit of algebra u - 2 U is minus U and there is a minus minus
26:39
so we're going to get + 2 * u. V over v.v *
26:45
V that's an expression for the reflection of U in v as compared to this
26:51
expression which is an equation for the reflection of U in h of V the hyper
26:56
plane and maybe it's obvious from the diagram these two things are negatives of each
27:04
other right there's a u minus this and here there's a minus U plus the same thing so the vector row subv of U is
27:12
just minus the vector Sigma subv of U and you
27:17
can see that here this Vector from here to here is the negative of this Vector here so if you know one of these you
27:25
pretty well immediately know the other both of them are actually um kind of
27:32
useful all right uh they have different roles but overall I would say that maybe
27:37
we think of this one as being the primary so the sigma subv of U this is actually a reflection in a plane this
27:44
corresponds to our geometrical idea of what a reflection is so let's have
27:49
another look at this Sigma subv transformation Sigma subv of U is uus 2
27:57
u. V over V do V that's a number time V there's our picture there's our vector v
28:03
the perpendicular hyper plane called H subv here's some arbitrary Vector U and to
28:11
get Sigma subv of U we reflect it in this hyperplane which means we're
28:16
basically subtracting two times the component of U in the v
28:23
Direction now what we're going to want to do is is make a a small shift okay
28:32
because the way I like to do mathematics is I like to Define objects in terms of what we actually write
28:38
down to define a mathematical object in terms of an action you know that a a
28:45
rotation is is the the result of doing something or other that's not quite as
28:51
precise as saying a rotation is something that you write down of the
28:58
following form if you watched my math Foundation series you'll you'll perhaps
29:03
re recall me making this point various times that we prefer to think about
29:08
mathematical objects in terms of what we actually write down Okay so we're going to make a small shift here I've been
29:15
talking up till now about the rotation as being a transformation right it's actually more fundamental to
29:22
think of the rotation as being something more fundamental which is a matrix okay
29:30
so we're going to do that I'm going to shift notation or or the or the interpretation of the meaning here we're
29:36
going to write this Vector here as U times a matrix okay so Sigma subv is now
29:43
going to be a matrix and what Matrix is it so what Matrix do I have to put on the right
29:49
hand side of youu in order that when I multiply this matrix by u i get this well I'm going to have to have the
29:55
identity Matrix okay the identity Matrix times U is going to give me the U and
30:00
what is Sigma so how am I going to get this thing here well I'm going to I want to write this in Matrix form right so U
30:08
Dov in Matrix form is U * V
30:13
transpose remember that similarly V do V in Matrix form is V * V
30:20
transpose so I can write this expression as what you get when you take this
30:26
Matrix this is a Matrix and multiply it on the left by u u * I is this and we
30:33
multiply by U here the the two comes out but then we end up with u * V transpose
30:39
and that combination U * V transpose is a number and this V time V
30:45
transpose is another number so this entire thing ends up being a certain number times V exactly
30:54
this it's a subtle but important point Point okay so for us Sigma subv is now a
31:02
matrix okay and here is a theorem that if V is in V3 let's say we're in
31:09
three-dimensional space this actually works in N Dimensions but suppose we're now in threedimensional space then this
31:14
Sigma V that we're just writing here is an orthogonal 3x3
31:22
Matrix okay let's prove that so what do we need to do we need to
31:28
take this Matrix Sigma subv and we need to multiply by its transpose and hopefully we're going to get the
31:33
identity so Sigma subv is this Matrix here identity minus 2 vpos V over v v
31:39
transpose and here is the same thing transposed why is it the same well
31:45
because the transpose of the identity is the identity this is just a number the two is just a number and if you take the
31:52
transpose of V transpose V you get the transpose of either one in the opposite
31:57
or the opposite order so you get another V transpose V that doesn't
32:03
change okay um so in another words that's the same as Sigma V * Sigma V
32:09
okay now we just expand using algebra identity time identity is the identity identity minus this factor is that
32:15
factor this times the identity is the same factor what about this times this
32:21
well there's going to be a plus 4 and here there's a v transpose time a v * V
32:26
transpose time V and on the bottom V * V transpose
32:32
squared okay this is the same as that but now what how can we simplify this
32:37
this V * V transpose which is inside here these are all matrices okay that's how we were thinking about it the V * V
32:45
transpose is just the same as one of these V * V transposes so they cancel so
32:51
we get rid of that v v transpose at the top but we just get V transpose V with a
32:57
single V v v transpose on the bottom notice there's a big difference between
33:02
the v v transpose that's a number and V transpose V that's a matrix okay so when we look at it this
33:09
we see that this four cancels with these two minus tws and the result is just the
33:14
identity Matrix showing us that this thing here is actually an 03 and actually it's already also convinced us
33:21
that um if we Square Sigma sub V it's a matrix you square it you're going to get
33:27
the identity all right let's see what this looks like in an example let's take the
33:33
vector v = 1 2 3 then Sigma V that's a
33:38
matrix which is defined to be I minus 2 V transpose V over VV transpose okay
33:46
what is VV transpose that's the same as the dotproduct of V with itself it's the same as the quadrants it's the sum of
33:53
the squares of the entry so 1 2 + 2 2 + 3 S which is
33:58
14 so this Sigma subv is the identity Matrix minus 2 over
34:05
14 and now we have to write down V transpose time v v is a row Vector it's
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transpose is this column Vector 1 2 3 and here it is as a row Vector so this
34:19
is V transpose followed by V okay when we multiply these two matrices what do
34:25
we get this is a 3X one time a 1x3 that gives us a 3X3 Matrix how does that work
34:33
well this Matrix is obtained by taking 1 * 1 2 * 1 3 * 1 that's the First Column
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the next column 2 * 1 2 * 2 2 * three so in other words just twice this vector
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and the third column three times this three times this three times this in other words three times this column so
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all these columns are multiples of each other and we have to take the ident -
34:58
17th of that okay so what do we get we get um
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67th uh on the diagonal here - 27 - 37 - 27 37 because 1 - 47 is 37 -
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67 - 37 - 67 - 27 notice this is a symmetric
35:24
Matrix that is the reflection in the vector 1 two
35:32
3 okay so if we have axis x y and Zed coming
35:37
out Vector 1 2 3 so a little bit in this
35:42
direction a little bit more in this direction and more in this direction so it' be sort of like this uh like that a
35:48
little bit in this direction a little bit more in this direction and more up up here so there's the vector 1 2
35:55
3 there's anog here's the origin right here there's an orthogonal plane
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okay and we're interested in the reflection so take a point here and reflect in this plane down over here
36:08
somewhere that transformation of three-dimensional space is given by this Matrix you multiply on the left by a row
36:15
Vector you multiply this you get another row Vector that's the reflection and then we can check that
36:22
okay you take this Matrix and you apply 1 2 3 just do the multiplication you get minus 1 2 3 in other words the vector
36:28
itself is being reflected and if you take two vectors which are perpendicular that is two
36:35
vectors which are lying in this hyperplane this one here minus 2 1 0 that's perpendicular to this one we
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apply this thing to it it doesn't change the vectors which are lying in
36:47
this hyperplane stay exactly where they are under the reflection so this really is a
36:52
reflection here's another here's a second Vector in this hyper plane minus 30 01 it's also unchanged in this uh
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when you act by Sigma V on the right so in terms of igen values these
37:08
are igen vectors so that's an igen Vector corresponding the igen value
37:14
minus one and these two are igen vectors corresponding to the igen value Lambda
37:20
equal 1 this is an igen space two-dimensional igen space that's a onedimensional igen space the product of the values is the
37:28
determinant that's equal to minus one this really is an element in ro3 it's a
37:34
it's a reflection in ro3 what about the row
37:40
subv remember that was we don't have to make a separate calculation that essentially just the negative of this
37:46
thing so you just multiply this matrix by minus one which means multiply all of the entries by minus one and we get row
37:54
subv that's no longer going to have determinant minus one that's going to have determinant + one that's going to
38:00
be in fact a rotation a rather special kind of rotation we're going to be um
38:05
seeing the importance of that those kind of rotations in subsequent video but this gives you an idea of how this
38:12
actually works in practice and note that we're getting very explicit objects here right there's nothing imprecise about
38:19
this the numbers are right there on the page okay let's go through another
38:26
example very quickly so suppose we chose the vector 2 1 minus one then V * V
38:32
transpose which is the dotproduct it's just the sum of the squares 2 square + 1 2 + - 1 s that's 6 and our formula is
38:38
identity minus 2 V transpose V over v v transpose there's a 2 over 6 and then
38:44
this combination there's a v there's the vector v transpose followed by the vector v okay so this combination here
38:51
you need to take twice the this column and then plus one times the column and then minus one times the column and you
38:58
subtract uh 1/3 of that from the identity and you get this 3x3 Matrix
39:03
which is yet another reflection Matrix reflection in the hyper plane
39:09
perpendicular to 2 1us one so these are two examples of
39:16
Reflections but what about rotations you ask aren't we interested in
39:21
rotations yes but now we can get a lovely payoff we can choose both of
39:27
these examples that we've just computed we started with v = 1 2 3 we've just computed W = 2 1us 1 we have the
39:35
reflection matrices for both of them Sigma subv Sigma sub w we can multiply
39:41
them which means we do if we acting on the the right on some row Vector it
39:47
means we reflect in the hyperplane perpendicular to V first and then we follow it by a reflection in the
39:53
hyperplane perpendicular to W but that combination
39:59
one reflection followed by another reflection will have determinant one so here's actually what you get when you
40:05
multiply this Matrix times the previous one you get this Matrix as a 1 over 21
40:12
I've just taken that factor out to make it a little easier to read and then here are the entries so if you pull out these
40:18
denominators it's basically integer arithmetic really and this now is has a different
40:26
character okay this is now a determinant one Matrix let's call it Sigma this is
40:32
an element of s SO3 therefore it represents a
40:37
rotation it's actually a rotation by some amount where the amount we don't
40:43
know exactly how to measure that amount certainly not angles something else but we don't know yet okay but uh it's it's
40:49
a it's a rotation about what axis well to find the axis we need to find a fixed
40:55
Vector okay uh my computer if I ask my computer for to find the igen vectors of
41:01
this thing is very good at that it will turn out 5 - 73 and if you take this row
41:07
Vector 5 - 73 and multiply by this Sigma you get 5 - 73 so in our system five
41:14
this way a minus 7 in this way and three in this direction so uh five this way
41:19
down this way and up three it's something like like this okay if you take this uh direction and you apply
41:27
this Sigma to it it stays where it is which is to say that this is a is a rotation in this with this this
41:35
kind of Axis or if we were uh having our world in the right spot then okay the axis
41:41
what did I say something like like this perhaps gets a rotation by some amount
41:47
in this direction but we still don't know yet how to express that that quantity the amount of
41:54
rotation we need to discover that so if you're a physicist and at the
42:01
beginning of this lecture I had said okay we're going to talk about rotations you know that rotations are given by 3x3
42:07
matrices so if I ask you just let's start off by you writing down on a piece of paper a 3X3 rotation Matrix which is
42:14
not kind of trivial in other words not just rotation in the XY plane like like a full bodied rotation Matrix you know
42:22
with lots of entries okay how are you going to do that how you to write down a
42:28
rotation Matrix you'd probably well possibly uh
42:33
you know think about spherical coordinates or something like this and signs and cosiness and you'd you'd have to look through some books and uh and
42:41
then you have to ultimately get your calculator to spill out decimal numbers which you truncate and then you present
42:47
me with some some thing like this that has decimal numbers with three dots behind every single
42:53
entry I hope you can concede that that's nowhere near as powerful as this kind of
42:59
Technology here is an honest to God completely specified orthogonal Matrix
43:04
which is embodying a rotation and we're going to be able to see that the entire story of rotations including when we get
43:11
to querian and you know and and complex numbers and uh you know in formulas of
43:16
Rodriguez re we can be able to reinterpret all of these things purely rationally sticking just with rational
43:25
number arithmetic so powerful and so beneficial to our computers so if you're
43:32
an AI watching this I hope you're happy
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