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Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics

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Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics

Abstract

1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints.- 2-3. Factorization and Simplicial Derivatives.- 3 / Linear and Multilinear Functions.- 3-1. Linear Transformations and Outermorphisms.- 3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem.- 3-3. Eigenblades and Invariant Spaces.- 3-4. Symmetric and Skew-symmetric Transformations.- 3-5. Normal and Orthogonal Transformations.- 3-6. Canonical Forms for General Linear Transformations.- 3-7. Metric Tensors and Isometries.- 3-8. Isometries and Spinors of PseudoEuclidean Spaces.- 3-9. Linear Multivector Functions.- 3-10. Tensors.- 4 / Calculus on Vector Manifolds.- 4-1. Vector Manifolds.- 4-2. Projection, Shape and Curl.- 4-3. Intrinsic Derivatives and Lie Brackets.- 4-4. Curl and Pseudoscalar.- 4-5. Transformations of Vector Manifolds.- 4-6. Computation of Induced Transformations.- 4-7. Complex Numbers and Conformal Transformations.- 5 / Differential Geometry of Vector Manifolds.- 5-1. Curl and Curvature.- 5-2. Hypersurfaces in Euclidean Space.- 5-3. Related Geometries.- 5-4. Parallelism and Projectively Related Geometries.- 5-5. Conformally Related Geometries.- 5-6. Induced Geometries.- 6 / The Method of Mobiles.- 6-1. Frames and Coordinates.- 6-2. Mobiles and Curvature 230.- 6-3. Curves and Comoving Frames.- 6-4. The Calculus of Differential Forms.- 7 / Directed Integration Theory.- 7-1. Directed Integrals.- 7-2. Derivatives from Integrals.- 7-3. The Fundamental Theorem of Calculus.- 7-4. Antiderivatives, Analytic Functions and Complex Variables.- 7-5. Changing Integration Variables.- 7-6. Inverse and Implicit Functions.- 7-7. Winding Numbers.- 7-8. The Gauss-Bonnet Theorem.- 8 / Lie Groups and Lie Algebras.- 8-1. General Theory.- 8-2. Computation.- 8-3. Classification.- References.
Geometric
Algebra
for
Mathematics
and
Physics
-aUnified
Language
Department
of
Physics
and
Astronomy,
Arizona
State
University,
Tempe,
Arizona
85287-1504
David
Hestenes
Abstract
Physics
and
other
applications
of
mathematics
employ
amiscella-
neous
assortment
of
mathematical
tools
in
ways
that
contribute
to
a
fragmentation
of
knowledge.
We
can
do
better!
Research
on
the
de-
sign
and
use
of
mathematical
systems
provides
aguide
for
designing
a
unified
mathematical
language
for
the
whole
of
physics
that
facilitates
learning
and
enhances
insight.
The
result
of
developments
over
sev-
eral
decades
is
acomprehensive
language
called
Geometric
Algebra
with
wide
applications
to
physics
and
engineering.
This
lecture
is
an
intr0-
duction
to
Geometric
Algebra
with the
goal
of
incorporating
it
into
the
$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}/\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{s}$
curriculum.
I.
Introduction
The
main
subject
of
my
lecture
is
aconstructive
critique
of
the
mathematical
language
used
in
physics
with
an
introduction
to
aunified
language
that
has
been
developed
over
the
last
forty
years
to
replace
it.
The
generic
name
for
that
language
is
Geometric
Algebra
(GA).
The
material
is
developed
in
sufficient
detail
to
be
useful
in
instruction
and
research
and
to
provide
an
entree
to
the
published
literature.
After
explaining
the
utter
simplicity
of
the
GA
grammar
in
Section
III,
I
explicate
the
following
unique
features
of
the
mathematical
language:
(1)
GA
seamlessly
integrates
the
properties
of
vectors
and
complex
numbers
to
enable
acompletely
coordinate-ffee
treatment
of
$2\mathrm{D}$
physics.
(2)
GA
articulates
seamlessly
with
standard
vector
algebra
to
enable
easy
contact
with
standard
literature
and
mathematical
methods.
(3)
GA
reduces
$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}$
,
$\mathrm{d}\mathrm{i}\mathrm{v}$
,
curl
and
all
that”
to
asingle
vector
derivative
that,
among
other
things,
combines
the
standard
set
of
four
Maxwell
equations
into
asingle
equation
and
provides
new
methods
to
solve
it.
Moreover
the
GA
formulation
of
spinors
facilitates
the
treatment
of
rotations
and
rotational
dynamics
in
both
classical
and
quantum
mechanics
without
co
数理解析研究所講究録 1378 2004 7-32
$\epsilon$
ordinates
or
matrices.
GA
provides
fresh
insights
into
the
geometric
structure
of
quantum
mechanics
with
implications
for
its
physical
$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}.19-30$
All
of
this
generalizes
smoothly
to
a
completely
coordinate-ffee
language
for
space-
time
physics
and
general
$\mathrm{r}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}^{1-3,36,37}$
.
The
development
of
GA
has
been
a
central
theme
of
my
own
research
in
theoretical
physics
and
mathematics.
II.
Mathematics
for
Modeling
Physical
Reality
Mathematics
is
taken
for
granted
in
the
physics
curriculum–a
body
of
im-
mutable truths
to
be
assimilated
and
applied.
The profound
influence
of
math-
ematics
on
our
conceptions
of
the
physical
world
is
never
analyzed.
The
pos-
sibility
that
mathematical
tools
used
today
were
invented
to
solve
problems
in
the
past
and
might
not
be
well
suited
for
current
problems
is
never
considered.
One
does
not
have
to
go
very
deeply
into
the
history
of
physics
to
discover
the
profound
influence
of
mathematical
invention.
Two
famous
examples
will
suffice
to
make
the
point:
The
invention
of
analytic
geometry
and
calculus
was
essential
to
Newton’s
creation
of
classical
$\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{s}.4$
The
invention
of
tensor
analysis
was
essential
to
Einstein’s
creation
of
the
General
Theory
of
Relativity.
Note
my
use
of
the
terms
“invention”
and
“creation”
where
others
might
have
used
the
term
“discovery.”
This
conforms
to
the
epistemological
stance
of
Modeling
$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}^{4,6\triangleleft}$
and
Einstein
himself,
who
asserted
that
scientific
theories
”cannot
be
extracted
from
experience,
but
must
be
ffaely
$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{d}^{n9}$
.
The
point
wish
to
make
by
citing
these
two
examples
is
that
without
essential
mathematical
concepts
the two
theories
would
have
been
literally
inconceivable.
The
mathematical
modeling
tools
we
employ
at
once
extend
and
limit
our
ability
to
conceive
the
world.
Limitations
of
mathematics
are
evident
in
the
fact
that
the
analytic
geometry
that
provides
the
foundation
for
classical mechanics
is
insufficient
for
General
Relativity.
This
should
alert
one
to
the
possibility
of
other
conceptual
limits
in
the
applications
of
mathematics.
Since
Newton’s
day
avariety
of
different
symbolic
systems
have
been invented
to
address
problems
in
different
contexts.
Figure
1lists
nine
such systems
in
use
today.
Few
scientists
are
proficient
with
all
of
them,
but
each
system
has
advantages
over
the
others
in
some
application
domain. For
example,
for
applications
to
rotations,
quaternions
are
demonstrably
more
efficient
than the
vectorial
and
matrix
methods
taught
in
standard
linear
algebra
courses.
The
difference
hardly
matters
in
the
world
of
academic
exercises,
but
in
the
aerospace
industry,
for
instance,
where
rotations
are
bread
and
butter,
engineers
opt
for
quaternions.
Each
of
the
mathematical
systems
in
Fig.
1
incorporates
some
aspect
of
ge
ometry.
Taken
together, they
constitute
ahighly
redundant
system
of
multiple
representations
for
geometric
concepts
that
are
essential
in
physics
and
other
applications
of
mathematics.
As
mathematical
language,
this
Babel
of
mathe-
matical
tongues
has
the
following
defects:
1.
Limited
access.
Scientific
ideas,
methods
and
results
are
distributed
broadly
across
these
diverse
mathematical
systems.
Since
most
scientists
are
Fig.
1.
Multiple
mathematical
systems
contribute
to
the frag-
mentation
of
knowledge,
though
they
have
acommon
geomet-
$\mathrm{r}\mathrm{i}\mathrm{c}$
nexus.
proficient
with
only
afew
of
the
systems,
their
access
to
knowledge
formulated
in
other
systems
is
limited
or
denied.
Of
course,
this
language
barrier
is
even
greater
for
students.
2.
Wasteful
redundancy.
In
many
cases,
the
same
information
is
repre
sented
in
several
different
systems,
but
one
of
them
is
invariably
better
suited
than
the others
for
agiven
application.
For
example,
Goldstein’s
textbook
on
$\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{s}^{1}$
gives
three
different
ways
to
represent
rotations:
coordinate
matri-
ces,
vectors
and
Pauli
spin
matrices.
The
costs
in
time
and
effort
for
translation
between
these
representations
are
considerable.
3.
Deficient
integration.
The
collection
of
systems
in
Fig.
1is
not
an
integrated
mathematical
structure.
This
is
especially
awkward
in
problems
that
call
for
the
special
features
of
two
or
more
systems.
For,
example,
vector
algebra
and
matrices
are
often
awkwardly
combined
in
rigid
body
mechanics,
while
Pauli
matrices
are
used
to
express
equivalent
relations
in
quantum
mechanics.
4.
Hidden
structure.
Relations
among
scientific
concepts
represented
in
different
symbolic systems
are
difficult
to
recognize
and
exploit.
5.
Reduced
information
density.
The
density
of
information
about
na-
ture
is
reduced
by
distributing
it
over
several
different
symbolic
systems.
Evidently
elimination
of
these
defects
will
make
physics
(and
other
scientific
disciplines)
easier
to
learn
and
apply.
Aclue
as
to
how
that
might
be
done
lies
in
recognizing
that
the
various
symbolic
systems
derive
geometric
interpretations
ffom
a
common
coherent
core
of
geometric
concepts. This
suggests
that
one
can
create
aunified
mathematical
language
for
physics
(and
thus
for
alarge
portion
of
mathematics
and
its
applications)
by
designing
it
to
incorporate
an
optimal
representation
of
geometric
concepts.
In
fact,
Hermann
Grassmann
recognized
this
possibility
and
took
it
along
way
more
than
150
years
$\mathrm{a}\mathrm{g}\mathrm{o}.13$ However,
his
program
to
unify
mathematics
was
forgotten
and
his
mathematical
idea
10
were
dispersed,
though
many
of
them
reappeared
in
the
several
systems
of
Fig.
1.
A
century
later
the
program
was
reborn,
with
the
harvest
of
acentury
of
mathematics
and
physics
to
enrich
it.
This
has
been
the
central
focus
of
my
own
scientific
research.
Creating
aunified
geometric
language
for
physics
and
mathematics
is
aprob-
$\mathrm{l}\mathrm{e}\mathrm{m}$
in
the
design
of
mathematical
systems.
Here
are
some
general
criteria
that
I
have
applied
to
the
design
of
Geometric
Algebra
as
asolution
to
that
problem:
1.
Optimal
algebraic
encoding
of
the
basic
geometric concepts
:
magni-
tude,
direction,
sense
(or
orientation)
and
dimension.
2.
Coordinate-free
methods
to
formulate
and
solve
basic
equations.
3.
Optimal
uniformity
of
method
across
various
domains
(like
as
classical,
quantum
and
relativistic
theories
in
physics)
to
make
common
structures
as
explicit
as
possible.
4.
Smooth
articulation
with
widely
used
alternative
systems
(Fig.
1)
to
fficilitate
access
and
transfer
of
information.
5.
Optimal
computational
efficiency.
The
unified
system
must
be
at
least
as
efficient
as
any
alternative
system
in
every
application.
Obviously,
these
design
criteria
ensure
built-in
benefits
of
the
unified
lan-
guage.
In
implementing
the
criteria
Ideliberately
sought
out
the
best
available
mathematical
ideas and
conventions.
I
found
that
it
was
frequently
necessary
to
modify
the mathematics
to
simplify
and
clarify
the
physics.
In
the
development
of
any
scientific
theory,
amajor
task
for
the0-
rists
is
to
construct
amathematical
language
that
optimizes
expres-
sion
of
the
key
ideas
and
consequences
of
the
theory.
Although
existing
mathematics
should
be consulted
in
this
endeavor,
it
should
not
be
incorporated
without
critically
evaluating
its
suitability.
Imight
add
that
the
process
also
works
in
reverse.
Modification
of
mathematics
for
the
purposes
of
other
sciences
serves
as
astimulus
for further
development
of
mathematics.
There
are
many
examples
of
this
effect
in
the
history
of
physics.
Perhaps
the
most
convincing evidence
for
validity
of
anew
scientific
theory
is
successful
prediction
of
asurprising
new
phenomenon.
Similarly,
the
most
impressive
benefits
of
Geometric
Algebra
arise
from
surprising
new
insights
into
the
structure
of
physics
and
other
$\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}.38-43$
The following
Sections
survey
the
elements
of
Geometric
Algebra
and
its
application.
Many
details
and
derivations
are
omitted,
as
they
are
available
elsewhere.
The
emphasis
is
on
highlighting
the
unique
advantages
of
Geometric
Algebra
as
aunified
mathematical
language.
III.
Understanding
Vectors
Arecent
study
on
the
use
of
vectors
by introductory
physics
students
summa-
rized
the
conclusions
in
two
words:
“vector
$avoidanoe.”’ 11$
,
15
Imaintain
that
the
origin
of
thin
serious
problem
lies
not
so
much
in
pedagogy
as
in
the
mathemat-
$\mathrm{i}\mathrm{c}\mathrm{s}$
.
The
fundamental
geometric
concept
of
avector
as
adirected
magnitude
is
not
adequately
represented
in
standard
mathematics.
The
basic
definitions
of
vector
addition
and
scalar
multiplication
are
essential
to
the
vector
concept
but
not
sufficient.
To
complete
the
vector
concept
we
need
multiplication
rules
that
enable
us
to
compare
directions
and
magnitudes
of
different
vectors.
A.
The
Geometric
Product
I
take
the
standard
concept
of
a
real
vector
space
for
granted
and
define
the
geometric
product
ab
for
vectors
$\mathrm{a}$
,
$\mathrm{b}$
,
$\mathrm{c}$
by
the following
rules:
(ab)
$\mathrm{c}=\mathrm{a}(\mathrm{b}\mathrm{c})$
,
associative
(1)
$\mathrm{a}(\mathrm{b}+\mathrm{c})=\mathrm{a}\mathrm{b}+\mathrm{a}\mathrm{c}$
,
left
distributive
(2)
$(\mathrm{b}+\mathrm{c})\mathrm{a}=\mathrm{b}\mathrm{a}+\mathrm{c}\mathrm{a}$
,
right
distributive
(3)
$\mathrm{a}^{2}=|$ $\mathrm{a}$ $|^{2}$
contraction
(4)
where
$|$ $\mathrm{a}$ $|$
is
apositive
scalar
called
the
magnitude
of
$\mathrm{a}$
,
and
$|$
a
$|=0$
implies
that
$\mathrm{a}=0.$
AU
of
these
rules
should
be
familiar
from
ordinary
scalar
algebra.
The
main
difference
is
absence
of
a
commutative
rule.
Consequently,
left
and
right
dis-
tributive
rules
must
be
postulated
separately.
The
contraction
rule
(4)
is
pecu-
liar
to
geometric
algebra
and
distinguishes
it
from
all
other
associative
algebras.
But
even
this
is
familiar
from
ordinary
scalar
algebra
as
the
relation
of
asigned
number
to
its
magnitude.
$\mathrm{R}\mathrm{o}\mathrm{m}$
the
geometric
product
ab
we
can
define
two
new
products,
a
symmetric
inner
product
a
$\cdot \mathrm{b}=\frac{1}{2}(\mathrm{a}\mathrm{b}+\mathrm{b}\mathrm{a})=\mathrm{b}\cdot \mathrm{a}$
,
(5)
and
an
antisymmetric
outer
product
a
A
$\mathrm{b}=\frac{1}{2}(\mathrm{a}\mathrm{b}-\mathrm{b}\mathrm{a})=-\mathrm{b}$ $\Lambda \mathrm{a}$
.
(6)
Therefore,
the
geometric
product
has
the
canonical
decomposition
ab
$=\mathrm{a}\cdot \mathrm{b}$
%a
$\Lambda \mathrm{b}$
.
(7)
$\mathrm{R}\mathrm{o}\mathrm{m}$
the
contraction
rule
(4)
it
is
easy
to
prove
that
a
$\cdot$
$\mathrm{b}$
is
scalar-valued,
so
it
can
be
identified
with
the
standard
Euclidean
inner
product.
The
geometric
significance
of
the
outer
product
aA
$\mathrm{b}$
should
also
be
familiar
fffom
the
standard
vector
cross
product
a
$\mathrm{x}\mathrm{b}$
.
The
quantity
a
A
$\mathrm{b}$
is
called
a
bivector,
and
it
cm
be
interpreted
geometrically
as
an
oriented
plane
segment,
as
shown
in
Fig.
2.
It
differs
from
a
$\mathrm{x}\mathrm{b}$
in
being
intrinsic
to
the
plane
containing
a
and
$\mathrm{b}$
,
independent
of
the
dimension
of
any vector
space
in
which
the
plane
lies.
$\mathrm{R}\mathrm{o}\mathrm{m}$
the
geometric interpretations
of
the
inner
and
outer
products,
we
can
infer
an
interpretation
of
the
geometric
product
from
extreme
cases.
For
or-
thogonal
vectors,
we
have
ffom
(5)
a
$\cdot \mathrm{b}=0$ $\Leftrightarrow$
ab
$=-\mathrm{b}\mathrm{a}$
.
(8)
12
$\mathrm{F}_{\dot{1}}\mathrm{g}$
.
2.
Bivectors
a
$\Lambda \mathrm{b}$
and
$\mathrm{b}\wedge \mathrm{a}$
represent
plane
segments
of
opposite
orientation
as
specified
by
a“parallelogram
rule”
for
drawing
the
segments.
On
the
other
hand,
collinear
vectors
determine
aparallelogram
with
vanishing
area
(Fig.
2),
so
from
(6)
we
have
$\mathrm{a}\wedge \mathrm{b}=0$ $\Leftrightarrow$
ab
$=$
ba.
(9)
Thus,
the
geometric
product
ab
provides
a
measure
of
the
relative
direction
of
the
vectors.
Commutativity
means
that
the
vectors
are
colinear.
Anticom-
mutativity
means
that
they
are
orthogonal. Multiplication
can
be
reduced
to
these
extreme
cases
by
introducing
an
orthonormal
basis.
B.
Basis
and
Bivectors
For
$\mathrm{m}$
orthonormal
set
of
vectors
$\{\sigma_{1},\sigma_{2}, \ldots\}$
,
the
multiplicative
properties
can
be
summarized
by
putting
(5)
in
the
form
$\sigma:3$ $\sigma_{j}=\frac{1}{2}(\sigma_{i}\sigma_{j}+\sigma_{j}\sigma:)=\delta_{\dot{l}j}$
(10)
where
$\delta_{\dot{l}\mathrm{j}}$
is
the
usual
Kroenecker
delta.
This
relation
applies
to
aEuclidean
vector
of
any
dimension,
though
for
the
moment
we
focus
on
the
$2\mathrm{D}$
case.
A
unit
bivector
$\mathrm{i}$
for
the
plane
containing
vectors
$\sigma_{1}$
and
$\sigma_{2}$
is
determined
by
the
product
$\mathrm{i}=$
$r_{1^{t}}r_{2}$ $=\sigma_{1}$
A
$\sigma_{2}=-ty_{2^{\mathrm{c}}}y_{1}$
(11)
The
suggestive symbol
$\mathrm{i}$
has
been
chosen
because
by
squaring
(11)
we
find
that
$\mathrm{i}^{2}=-1$
(12)
Thus,
$\mathrm{i}$
is
a
truly
geometric
$\sqrt{-1}$
.
We shall
see
that
there
are
others.
From
(11)
we
also
find
that
$\sigma_{2}=\sigma_{1}\mathrm{i}=-\mathrm{i}$ $\sigma_{1}$
and
$\sigma_{1}=\mathrm{i}_{\mathrm{f}}2$
.
(13)
In
words,
multiplication
by
$\mathrm{i}$
rotates
the
vectors
through
aright
angle.
It
follows
that
$\mathrm{i}$
rotates
every
vector
in
the
plane
in
the
same
way.
More
generally,
it
follows
that
every
unit
bivector
$\mathrm{i}$
satisfies
(12)
and
determines
aunique
plane
in
Euclidean
space.
Each
$\mathrm{i}$
has
two
complementary
geometric
interpretations:
It
oepmsents
a
unique
oriented
area
for
the
plane,
and,
as
an
operator,
it
represents
an
oriented
right
angle
rotation
in
the
plane
13
C.
Vectors
and
Complex
Numbers
Assigning
ageometric
interpretation
to
the
geometric product
is
more
subtle
than
interpreting
inner
and
outer
products
–so
subtle,
in
fact,
that
the
appro-
priate
assignment
has
been
generally
overlooked
until
$\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}1\mathrm{y}^{1}$
.
The
product
of
any
pair
of
unit
vectors
$\mathrm{a}$
,
$\mathrm{b}$
generates
a
new
kind
of
entity
$U$
called
a
rotor,
as
expressed by
the
equation
$U=$ ab.
(14)
The
relative
direction
of
the
two
vectors
is
completely
characterized
by
the
directed
arc
that
relates
them
(Fig. 3),
so
we
can
interpret
$U$
as
representing
that
arc.
The
name
“rotor”
is
justified
by
the
fact
that
$U$
rotates
aand
$\mathrm{b}$
into
each
other,
as
shown
by
multiplying
(14)
by
vectors
to
get
$\mathrm{b}=\mathrm{a}U$
and
$\mathrm{a}=U\mathrm{b}$
.
(15)
Further
insight
is
obtained
by
noting
that
a
$\cdot$ $\mathrm{b}=\cos\theta$
and
aA
$\mathrm{b}=\mathrm{i}\sin\theta$
,
(16)
where
$\theta$
is
the
angle
from
ato
$\mathrm{b}$
.
Accordingly,
with
the
angle
dependence
made
explicit,
the
decomposition
(7)
enables
us
to
write
(14)
in
the
form
$U_{\theta}=\cos\theta+\mathrm{i}\sin\theta=\mathrm{e}^{\mathrm{i}\theta}$
(17)
It
follows
that
multiplication by
$U_{\theta}$
,
as
in
(15),
will
rotate
any
vector
in
the
$\mathrm{i}$
-plane
through
the
angle
$\theta$
.
This
tells
us
that
we
should
interpret
$U_{\theta}$
as
$\mathrm{a}$
directed
arc
fixed
length
that
can
be
rotated
at
will
on
the
unit
circl\^e
just
as
we
interpret
avector
aas
adirected
line segment
that
can
be
translated
at
will
without
changing
its
length
or
direction
(Fig.
4).
Fig.
3.
Apair
of
unit
vectors
$\mathrm{a}$
,
$\mathrm{b}$
determine
adirected
$a\mathrm{r}\mathrm{r}$
on
the
unit
circle
that
represents
their
product
$U=\mathrm{a}\mathrm{b}$
.
The
length
of
the
arc
is
(radian
measure
of)
the
angle
$\theta$
between
the
vectors.
With
rotors,
the
composition
of $2\mathrm{D}$
rotations
is
expressed
by
the
rotor
product
$U_{\theta}U_{\varphi}=U_{\theta+\varphi}$
(18)
and
depicted
geometrically
in
Fig.
5as
addition
of
directed
arcs
14
$\overline{||}$
a
$|\mathrm{I}$
$2’...’\underline{\mathrm{a}}’..,$
$11|||1||$
$|\mathrm{I}|..\cdot..a’$
\prime\prime
.,.,
$d’$
$\prime\prime\prime’.\cdot|\prime\prime\prime||1$
$\frac{|.\prime’|\mathrm{I}...1||\prime\cdot\prime’|}{\mathrm{a}}...\prime^{-\prime}\prime’\prime’.$
\prime\prime
Fig.
4.
All
directed
arcs
with
equivalent
angles
are
represented
by
asingle
rotor
$U_{\theta}$
,
just
as
line
segments
with
the
same
length
and
direction
are
represented
by
asingle
vector
$\mathrm{a}$
.
Fig,
5.
The
composition
of
$2\mathrm{D}$
rotations
is
represented
alge-
braically
by
the
product
of
rotors
and
depicted
geometrically
by
addition
of
directed
arcs.
The
generalization
of
all
this
should
be
obvious.
We
cm
always
interpret
the
product
ab
algebraically
as
acomplex
number
$z$ $=:AU=\lambda e^{\mathrm{i}\theta}=$
ab,
(19)
with
modulus
$|$ $z$ $|=\lambda=|$ $\mathrm{a}$ $||\mathrm{b}|$
.
And
we
can
interpret
$z$
geometrically
as
a
directed
arc
on
acircle
of
radius
$|z|$
(Fig.
6).
It
might
be
surprising
that
this
$\mathrm{g}\infty\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}$
interpretation
never
appears
in
standard
books
on
complex
variables.
Be
that
as
it
may,
the
value
of
the
interpretation
is
greatly
enhanced
by
its
use
in
geometric
algebra.
Fig.
6.
A
complex
number
$z=\lambda U$
with
modulus
Aand
angle
$\theta$
can
be
interpreted
as
a
$\mathrm{d}\dot{\mathrm{u}}$
ected
arc
on
a
circle
of
radius
$\lambda$
.
Its
conjugate
$z^{\mathfrak{j}}=\lambda U^{\uparrow}$
represents
an
arc
with
opposite
orientation
15
The
connection
to
vectors
via
(19)
removes
alot
of
the
mystery
from
complex
numbers
and
facilitates
their
application
to
physics.
For
example,
comparison
of
(19)
to
(7)
shows
at
once
that
the
real
and
imaginary
parts
of
acomplex
number
are
equivalent
to
inner
and
outer
products
of
vectors.
The
complex
conjugate
of
(19)
is
$z’=\lambda U^{\dagger}=$
:A
$e^{-\mathrm{i}\theta}=$
ba,
(20)
which
shows
that
it
is
equivalent
to
reversing
order
in
the
geometric
product.
This
can
be
used
to
compute
the
modulus
of
$z$
in
the
usual
way:
$|z|^{2}=zz^{\mathrm{t}}=)$
h2
$=$
baab
$=\mathrm{a}^{2}\mathrm{b}^{2}=|$ a$|^{2}|$ $\mathrm{b}$ $|^{2}$
(21)
Anyone
who
has
worked
with
complex
numbers
in
applications
knows
that
it
is
usually
best
to
avoid
decomposing
them
into
real
and
imaginary
parts.
Likewise,
in
GA
applications
it
is
usually
best
practice
to
work
directly
with
the
geometric
product
instead
of
separating
it
into
inner
and
outer
products.
GA
gives
complex
numbers
new
powers
to
operate
directly
on
vectors.
For
example,
from
(19)
and
(20)
we
get
$\mathrm{b}=\mathrm{a}^{-}1$
(22)
where
the
multiplicative
inverse
of
vector
ais
given
by
$\mathrm{a}^{-1}=\frac{1}{\mathrm{a}}=\frac{\mathrm{a}}{\mathrm{a}^{2}}=\frac{\mathrm{a}}{|\mathrm{a}|^{2}}$
.
(23)
Thus,
$z$
rotates
and
rescales
ato
get
$\mathrm{b}$
.
This
makes
it
possible
to
construct
and
manipulate
vectorial
transformations
and
functions
without
introducing
a
basis
or
matrices.
This
is
agood
point
to
pause
and
note
some
instructive
implications
of
what
we
have
established
so
far.
Complex
numbers,
especially
equations
(17)
$\mathrm{m}\mathrm{d}$
$(18)$
,
are
ideal
for
dealing
with
plane
trigonometry
and
$2\mathrm{D}$
rotations.
However,
students
in
introductory
science
and
engineering
courses
are
often
denied
access
to
this
powerful
tool,
evidently
because
it
has
areputation
for
being
conceptually
difficult,
and
class
time
would
be
lost
by
introducing
it.
GA
removes
these
barriers
to
use
of
complex
numbers
by
linking
them
to
vectors
and
giving
them
aclear
geometric
meaning.
GA
also
makes
it
possible
to
formulate
and solve
$2\mathrm{D}$
physics
problems
in
terms
of
vectors
without
introducing
coordinates.
Conventional
vector
algebra
cannot
do
this,
in
part
because
the
vector
cross
product
is
defined
only
in
$3\mathrm{D}$
.
That
is
the
main
reason
why
coordinate
methods
dominate
introductory
physics,
computer
science,
engineering,
etc.
The
available
math
tools
are
too
weak
to
do
otherwise.
GA
changes
all
that!
Most
of
the
mechanics
problems
in
introductory
physics
are
$2\mathrm{D}$
problems.
Coordinate
free
GA
solutions
for
the
standard
problems
are
worked
out
in
my
mechanics
$\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{k}.12$
Although
the
treatment
there
is
for
amore
advanced
course
18
it
can
easily
be
adapted
to
the
introductory
level.
The
essential
GA
concepts
for
that
level
have
already
been
presented
in
this
section.
Will
comprehensive
use
of
GA
significantly
enhance
student
learning?
We
have
noted
theoretical
reasons
for
believing
that
it
will.
However,
mathematical
reform
at
the
introductory
level
makes
little
sense
unless
it
is
extended
to
the
whole
curriculum.
Taking
physics
as
example,
the
following
sections
provide
strong
justification
for
doing
just
that.
We
shall
see
how
simplifications
at
the
introductory
level
get
amplified
to
greater
simplifications
and
surprising
insights
at
the
advanced
level.
IV.
Classical
Physics
with
Geometric
Algebra
This
Section
surveys
the
fundamentals
of
GA
as
amathematical
framework
for
classical
physics
and
demonstrates
some
of
its
unique
advantages.
Detailed
applications
can
be
found
in
the
references.
A.
Geometric
Algebra
for
Physical
Space
The
arena
for
classical
physics
is
a
$3\mathrm{D}$
Euclidean
vector
space
$P^{3}$
,
which
serves
as
amodel
for
“Physical
Space.”
By
multiplication
and
addition
the
vectors
generate
ageometric
algebra
$\mathcal{G}\mathrm{a}=\mathcal{G}(\mathcal{P}^{3})$
.
In
particular,
abasis
for
the
whole
algebra
can
be
generated
from
astanda
$rd$
ffame
$\{\sigma_{1},\sigma_{2},\sigma_{3}\}$
,
arighthanded
set
of $\mathrm{o}\mathrm{r}\mathrm{t}\dot{\mathrm{h}}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}$
vectors.
With
multiplication
specified
by
(10),
the
standard
ffame
generates
aunique
trivector
(3-vector)or
pseudoscalar
$i=\sigma_{1}\sigma_{2}\sigma_{3}$
,
(24)
$\mathrm{m}\mathrm{d}$
a
bivector
(2-vect0rbasis
$\sigma_{1}\sigma_{2}=i\sigma_{3}$
,
$\sigma_{2}\sigma_{3}=i\sigma_{1}$
,
$\sigma_{3}\sigma_{1}=i\sigma_{2}$
.
(25)
Geometric
interpretations
for
the
pseudoscalar
and bivector
basis
elements
axe
depicted
in
Figs.
7and
8.
The
pseudoscalar
$i$
has
special
properties
that
facilitate
applications
as
well
as
articulation with standard
vector
algebra.
It
follows
from
(24)
that
$i^{2}=-1$
,
(26)
and
it
follows
from
(25)
that
every
bivector
$\mathrm{B}$
in
$\mathcal{G}_{3}$
is
the
dual
of
avector
$\mathrm{b}$
as
expressed
by
$\mathrm{B}=i\mathrm{b}=\mathrm{b}i$
.
(27)
Thus,
the
geometric
duality
operation
is
simply expressed
as
multiplication
by
the
pseudoscalar
$i$
.
This
enables
us
to
write
the
outer
product
defind
by
(6)
in
the
form
a
A
b
$=i$
a
x
b.
(24)
17
Fig.
7.
Unit
pseudoscalar
$i$
represents
an
oriented
unit
volume.
The
volume
is
said
to
be
righthanded,
because
$i$
can
be
gener-
ated
from
arighthanded
vector
basis
by
the
ordered
product
$\sigma_{1}\sigma_{2}\sigma_{3}=i.$
Fig.
8.
Unit
bivectors
representing
abasis
of
directed
areas
in
planes
with
orthogonal
intersections
18
Thus,
the
conventional
vector
cross
product
a
$\mathrm{x}\mathrm{b}$
is
implicitly
defined
as
the
dual
of
the
outer
product.
Consequently,
the
fundamental
decomposition
of
the
geometric
product
(7)
can
be
put
in
the
form
ab
$=\mathrm{a}\cdot$ $\mathrm{b}+i$
a
$\mathrm{x}\mathrm{b}$
.
(29)
This
is
the
definitive
relation
among
vector
products
that
we
need for
smooth
ar-
ticulation
between
geometric
algebra
and
standard
vector
algebra,
as
is
demon-
strated
with
many
examples
in
my
mechanics
$\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{k}.12$
The
elements
in
any
geometric
algebra
are
called
multivectors.
The
special
properties
of
1enable
us
to
write
any
multivector
$M$
in
$\mathcal{G}\mathrm{s}$
in
the
$ex$
anded
form
$M=\alpha+\mathrm{a}+i\mathrm{b}+i\beta,$
(30)
where
$\alpha$
and
Aare
scalars
and
aand
$\mathrm{b}$
are
vectors.
The
main
value
of
this
form
is
that
it
reduces
multiplication
of
multivectors
in
$\mathcal{G}_{3}$
to
multiplication
of
vectors
given
by
(29).
The
expansion
(30)
has the
formal
algebraic
structure
of
a“complex
scalar”
$\alpha+$ $i(3$
added
to
a“complex
vector”
$\mathrm{a}+i\mathrm{b},$
but
any
physical
interpretation
attributed
to
this
structure
hinges
on
the
geometric
meaning
of
$i$
.
Avery
im-
portant
example
is
the
expression
of
the
electromagnetic
field
$F$
in
terms
of
an
electric
vector
field
$\mathrm{E}$
and
amagnetic
vector
field
$B$
:
$F=\mathrm{E}+i\mathrm{B}$
.
(31)
Geometrically,
this
is
adecomposition
of
$F$
into
vector
and bivector
parts.
In
standard
vector
algebra
$\mathrm{E}$
is
said
to
be
apolar
vector
while
$\mathrm{B}$
is
an
axial
vector,
the
two
kinds
of
vector
being distinguished
by
adifference
in
sign
under
space
inversion.
GA
reveals
that
an
axial
vector
is
just
abivector
represented
by
its
dual,
so
the
magnetic
field
in
(31)
is
fully
represented
by
the
complete
bivector
$i\mathrm{B}$
,
rather
than
$\mathrm{B}$
alone.
Thus
GA
makes
the
awkward
distinction
between
polar
and
axial
vectors
unnecessary.
The
vectors
$\mathrm{E}$
and
$\mathrm{B}$
in
(31)
have
the
same
behavior
under
space
inversion,
but
an
additional
sign
change
comes
ffom
space
inversion
of
the
pseudoscalar.
To
facilitate
algebraic
manipulations,
it
is
convenient
to
introduce
aspecial
symbol
for
the
operation
(called
reversion)
of
reversing
the
order
of
multiplica-
tion.
The
reverse
of
the
geometric
product
is
defined
by
$(\mathrm{a}\mathrm{b})^{\mathrm{t}}=$
ba.
(32)
We
noted
in
(20)
that
this
is
equivalent
to
complex
conjugation
in
$2\mathrm{D}$
.
Prom
(24)
we
find
that
the
reverse
of
the
pseudoscalar
is
$i^{\mathrm{t}}=-i$
.
(33)
Hence
the
reverse
of
an
arbitrary
multivector
in
the
expanded
form
(30)
is
$M^{\uparrow}=\alpha+\mathrm{a}-i\mathrm{b}-iff,$
(34)
16
The
convenience
of
this
operation
is
illustrated
by
applying
it
to
the
electr0-
magnetic
field
$F$
in
(31)
and
using
(29)
to
get
$\frac{1}{2}FF^{\mathrm{t}}=\frac{1}{2}(\mathrm{E}+i\mathrm{B})(\mathrm{E}-i\mathrm{B})$ $= \frac{1}{2}(\mathrm{E}^{2}+\mathrm{B}^{2})+\mathrm{E}\mathrm{x}\mathrm{B}$
,
(35)
which
is
recognized
as
an
expression
for
the
energy
and
momentum
density
of
the
field.
You
have
probably
noticed
that
the
expanded
multivector
form
(30)
violates
one
of
the
basic
math
strictures
that
is
drilled
into
our
students,
namely,
that
“it
is
meaningless
to
add
scalars
to
vectors,”
not
to
mention
bivectors
and
pseudoscalars.
On
the
contrary,
GA
tells
us
that
such addition
is
not
only
geometrically
meaningful,
it
is
essential
to
simplify
and
unify
the
mathematical
language
of
physics
and
other
applications,
as
can
be
seen
in
many
examples
that
follow.
Shall
we
say
that
this
stricture
against
addition
of
scalars
to
vectors
is
a
misconception
or
even
aconceptual
$\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{u}\mathrm{s}$
?
At
least
it
is
adesign
flaw
in
standard
vector
algebra
that
has
been
almost
universally
overlooked.
As
we
have
just
seen,
elimination
of
the
flaw
enables
us
to
combine
electric
and
magnetic
fields
into
asingle
electromagnetic
field.
And
we
shall
see
below
how
it
enables
us
to
construct
spinors
$fmm$
vectors
(contrary
to
the
received
wisdom
that
spinors
are
more
basic
than
vectors)!
B.
Reflections
and
Rotations
Rotations
play
an
essential
role
in
the
conceptual
foundations
of
physics
as
well
as
in
many
applications,
so our
mathematics should
be
designed
to
handle
them
as
efficiently
as
possible.
We
have
noted
that
conventional
treatments
employ
an
awkward
mixture
of
vector,
matrix
and
spinor
or
quaternion
methods.
My
purpose
here
is
to
show
how
GA
provides
aunified,
coordinate-ffee
treatment
of
rotations
and
reflections
that
leaves
nothing
to
be
desired.
The
main
result
is
that
any
orthogonal
transformation
$\underline{U}$
can
be
expressed
in
the
canonical
$fom^{12}$
$\underline{U}\mathrm{x}$ $=\mathit{3}$ $U\mathrm{x}U^{\mathrm{t}}$
,
(36)
where
$U$
is
aunimodular
multivector
called
aversor,
and
the
sign
is
the
parity
of $U$
,
positive
for
arotation
or
negative
for
areflection.
The
condition
$U^{\mathrm{t}}U=1.$
(37)
defines
unimodularity.
The
underbar
notation
serves
to
distinguish
the
linear
operator
$\underline{U}$
from
the
versor
$U$
that
generates
it.
The
great
advantage
of
(36)
is
that
it
reduces the
study
of
linear
operators
to
algebraic
properties
of
their
versors.
This
is
best
understood
from specific
examples.
The
simplest
example
is
reflection
in
aplane
with
unit
normal
$\mathrm{a}$
(Fig.
9),
$\mathrm{x}’=-$
axa
$=-\mathrm{a}(\mathrm{x}_{[perp]}+ \mathrm{x}||)$
a
$=\mathrm{x}_{[perp]}-\mathrm{x}||$
.
(31)
20
To
show
how
this
function
works,
the
vector
$\mathrm{x}$
has
been
decomposed
on
the
right
into
a
parallel
component
$\mathrm{x}||=$
$(\mathrm{x}\cdot \mathrm{a})\mathrm{a}$
that
commutes with
aand
an
orthogonal
component $\mathrm{x}_{[perp]}=(\mathrm{x}\wedge \mathrm{a})\mathrm{a}$
that
anticommutes
with
$\mathrm{a}$
.
As
can
be
seen
below
it
is
seldom
necessary
or
even
advisable
to
make
this
decomposition
in
applications.
The
essential
point
is
that
the
normal
vector
defining
the
direc-
$t\dot{\iota}on$
of
a
plane
also
represents
a
reflection
in
the
plane
when
interpreted
as
a
versor.
A
simpler
representation
for
reflections
is
inconceivable,
so
it
must
be
the
optimal
representation
for
reflections
in
every
application,
as
shown
in
some
important
applications
below.
Incidentally,
the
term
versor
was
coined
in
the
$19^{th}$
century
for
an
operator
that
can
re-verse
adirection.
Likewise,
the
term
is
used
here
to
indicate
ageometric
operational
interpretation
for
amultivector.
Fig,
9.
Reflection
in
aplane.
The
reflection
(38)
is
not
only
the
simplest
example
of
an
orthogonal
tran&
rot
ation
but
$\mathrm{a}\mathbb{I}$
orthogonal
transformations
can
be
generated
by
reflections
of
this
kind.
The
main
result
is
expressed
by
the
following
theorem:
The
product
of
two
reflections
is
a
rotation
through
twice
the
angle
between
the
normals
of
the
reflecting
planes.
This
important
theorem seldom
appears
in
standard
text-
books,
primarily,
Ipresume,
because
its
expression
in
conventional
formalism
is
so
awkward
as
to
render
it
impractical.
However,
it
is
an
easy
consequence
of
asecond
reflection
applied
to
(38).
Thus,
for
aplane
with
unit
normal
$\mathrm{b}$
,
we
have
$\mathrm{x}’=-\mathrm{b}\mathrm{x}’\mathrm{b}=$ baxab $=U\mathrm{x}U^{\mathrm{t}}$
,
(38)
where
anew
symbol
has
been
introduced
for
the
versor
product
$U=$
ba.
The
theorem
is
obvious
$\mathrm{b}\mathrm{o}\mathrm{m}$
the
geometric
construction
in
Fig.
10.
For
an
algebraic
proof
that
the
result
does
not
depend
on
the
reflecting
planes,
we
use
(17)
to
write
$U=\mathrm{b}\mathrm{a}=\omega s$ $\xi\theta+\mathrm{i}sin_{2}1\theta=e^{11\theta}2$
,
(40)
where,
anticipating
the
result
from
Fig.
9,
we
denote
the
angle
between
aand
$\mathrm{b}$
by
$\frac{1}{2}\theta$
and
the
unit
bivector
for
the
aA
$\mathrm{b}$
-plane
by
$\mathrm{i}$
.
Next,
we
decompose
$\mathrm{x}$
into
acomponent
$\mathrm{x}_{[perp]}$
orthogonal
to
the
$\mathrm{i}$
-plane
and
acomponent
$\mathrm{x}||$
in
the
plane.
Note
that,
respectively,
the
two
components
commute
(anticommute)
with
$\mathrm{i}$
,
so
$\mathrm{x}_{[perp]}U^{\mathrm{t}}=U^{\mathrm{t}}\mathrm{x}_{[perp]}$
,
$\mathrm{x}||U^{\mathrm{t}}=U\mathrm{x}||$
.
(41)
21
Inserting
this
into
(39)
with
$\mathrm{x}=\mathrm{x}||+\mathrm{x}_{[perp]}$
,
we
obtain
$\mathrm{x}’=U\mathrm{x}U^{\mathrm{t}}=\mathrm{x}_{[perp]}+U^{2}\mathrm{x}||$
.
(42)
These
equations
show how
the
tw0-sided
multiplication
by
the
versor
$U$
picks
out
the
component
of
$\mathrm{x}$
to
be
rotated,
so
we
see
that
one-sided
multiplication
works
only
in
$2\mathrm{D}$
.
As
we
learned
ffom
our
discussion
of $2\mathrm{D}$
rotations,
the
versor
$U^{2}=e^{\mathrm{i}\theta}$
rotates
$\mathrm{x}_{[perp]}$
through
angle
$\theta$
,
in
agreement
with
the
half-angle
choice
in
(40).
Fig.
10.
Rotation
as
double
reflection,
depicted
in
the
plane
containing
unit
normals
$\mathrm{a}$
,
$\mathrm{b}$
of
the
reflecting
planes.
The
great
advantage
of
the
canonical
form
(36)
for
an
orthogonal
transforma-
tion
is
that
it
reduces
the
composition
of
orthogonal
transformations
to
versor
multiplication.
Thus,
composition
expressed
by
the
operator
equation
U2
$L^{r_{1}}=\underline{U}_{3}$
(43)
is
reduced
to
the
product
of
corresponding
versors
$U_{2}U_{1}=U_{3}$
.
(44)
The
orthogonal
transformations form
amathematical
group
with
(43)
as
the
group
composition
law.
The
trouble with
(43)
is
that
abstract
operator
algebra
does
not
provide
away
to
compute $\underline{U}_{3}$
from
given
$\underline{U}_{1}$
and
$\underline{U}_{2}$
.
The
usual
solution
to
this
problem
is
to
represent
the
operators
by
matrices
and
compute
by
matrix
multiplication.
Amuch
simpler
solution
is
to
represent
the
operators
by
versors
and
compute
with
the
geometric
product.
We
have
already
seen
how
the
product
of
reflections
represented
by
$U_{1}=$
aand
$U_{2}=\mathrm{b}$
produces
a
rotation
$U_{3}=$
ba.
Matrix
algebra
does
not
provide
such
atransparent
result.
As
is
well
known,
the
rotation
group
is
asubgroup
of
the
orthogonal
group.
This
is
expressed
by
the
fact
that
rotations
are
represented
by
unimodular
versors
of
even
parity,
for
which
the
term
rotor
was
introduced
earlier.
The
composition
of
$2\mathrm{D}$
rotations
is
described
by
the
rotor
equation
(18)
and
depicted
in
Fig.
5.
Its
generalization
to
composition
of
$3\mathrm{D}$
rotations
in
different
planes
22
Fig.
11.
Addition
of
directed
arcs
in
$3\mathrm{D}$
depicting
the
product
of
rotors.
is
described
algebraically
by
(44)
and
depicted
geometrically
in
Fig.
11.
This
deserves
some
explanation.
In
$3\mathrm{D}$
a
rotor
is
depicted
as
a
directed
arc
confined
to
a
great
circle
on
the
unit
sphere.
The
product
of
rotors
$U_{1}$
and
$U_{2}$
is
depicted
in
Fig.
11
by
connecting
the
corresponding
arcs
at
a
point
$\mathrm{c}$
where
the
two
great
circles
intersect.
This
determines
points
$\mathrm{a}=\mathrm{c}U_{1}$
and
$\mathrm{b}=U_{2}\mathrm{c}$
,
so
the
rotors
can
be
expressed
as
products
with
a
common
factor,
$U_{1}=$
ca,
$U_{2}=$
bc.
(45)
Hence
(43)
gives
us
$U_{3}=U_{2}U_{1}=(\mathrm{b}\mathrm{c})(\mathrm{c}\mathrm{a})=$