[ Pobierz całość w formacie PDF ]

7
MULTIPLYING VECTORS
In your mathematical training so far, you will have various
products for vectors:
The Scalar Product
The scalar, (or inner or dot) product, returns a scalar from
two vectors. In Euclidean space the inner product is positive
definite,
From this we recover Schwarz inequality
We use this to define the cosine of the angle between and
via
Can now do Euclidean geometry. In non-Euclidean spaces,
such as Minkowski spacetime, Schwarz inequality does not
hold. Can still introduce an orthonormal frame. Some vectors
have squavre and some .
8
COMPLEX NUMBERS
A complex number
defines a point on an
Argand diagram. Com-
plex arithmetic is a way
of multiplying together
vectors in 2-d.
If then get length from
Include a second , and form
The real part is the scalar product. For imaginary term use
polar representation
Imaginary part is . The area of the
parallelogram with sides and . Sign is related to
handedness. Second interpretation for complex addition: a
sum between scalars and plane segments.
9
QUATERNIONS
Quaternion algebra contains 4 objects, , (instead
of 3). Algebra defined by
Define a closed algebra. (Also a division algebra  not so
important). Revolutionary idea: elements anticommute
Problem: Where are the vectors? Hamilton used  pure
quaternions  no real part. Gives us a new product:
Result of product is
is (minus) the scalar product. Vector term is
Defines the cross product . Perpendicular to the plane
of and , magnitude , and , and form a
right-handed set. The cross product was widely adopted.
10
THE OUTER PRODUCT
The cross product only exists in 3 dimensions. In 2-d there is
nowhere else to go, in 4-d the definition is not unique. In the
set any combination of and is perpendicular to
and .
Need a means of encoding a plane directly. This is what
Grassmann provided. Define the outer or wedge product
as directed area swept out by and . Plane has area
, defined to be the magnitude of .
Defines an oriented plane.
Think of as the parallelogram formed by sweeping one
vector along the other. Changing the order reverses the
orientation. Result is neither a scalar nor a vector. It is a
bivector  anew mathematical entity encoding the notion of a
plane.
11
PROPERTIES
1. The outer product of two vectors is antisymmetric,
This follows from the geometric definition. NB.
2. Bivectors form a linear space, the same way that vectors
do. In 3-d the addition of bivectors is easy to visualise. Not
always so obvious in higher dimensions.
3. The outer product is distributive
This helps to visualise the addition of bivectors.
12
4. The outer product does not retain information about shape.
If , have
Get same result, so cannot recover and from .
Sometimes better to replace the directed parallelogram with a
directed circle.
EXAMPLE  2 DIMENSIONS
Suppose are basis vectors and have
The outer product of these is
Same as imaginary term in the complex product . In
general, components are .
13
THE GEOMETRIC PRODUCT
Complex arithmetic suggests that we should combine the
scalar and outer products into a single product. This is what
Clifford did. He introduced the geometric product, written
simply as , and satisfying
Think of the right-hand side as like a complex number, with
real and imaginary parts, carried round in a single entity.
From the symmetry/antisymmetry of the terms on the
right-hand side, we see that
It follows that
Can define the other products in terms of the geometric
product. So treat the geometric product as the primitive one
and should define axioms for it. Properties of the other
products then follow.
14
GEOMETRIC ALGEBRA IN 2-D
Consider a 2-d space (a plane) spanned by 2 orthonormal
vectors ,
NB writing vectors in a bold face now!
The final entity present in the 2-d algebra is the bivector
. The highest grade element in the algebra, often
called the pseudoscalar (or directed volume element). Chosen
to be right-handed, so that sweeps onto in a
right-handed sense (when viewed from above). Use the
symbol for pseudoscalar
The full algebra is spanned by
1 scalar 2 vectors 1 bivector
Denote this algebra by . To study properties of first
form
For orthogonal vectors the geometric product is a pure
15
bivector. Also note that
so orthogonal vectors anticommute.
Now form products involving . Multiplying vectors
from the left,
o
A rotation clockwise (i.e. in a negative sense).
From the right
o
a rotation anticlockwise  a positive sense.
Finally form the square of ,
Have discovered a geometric quantity which squares to !
Fits with the fact that 2 successive left (or right) multiplications
o,
of a vector by rotates the vector through equivalent to
multiplying by .
16 [ Pobierz całość w formacie PDF ]