Grassmann Algebra

TheComplement.nb 32
hence can be factored into the exterior product of two vectors. It is in this form that the bivector
will be most easily interpreted as a geometric entity. There is an infinity of vectors that are
orthogonal to the vector x, but they are all contained in the bivector x
����� .
We can use the GrassmannAlgebra function FindFactoredForm discussed in Chapter 3 to
give us one possible factorization.
FindFactoredForm�a e2 � e3 � be1 � e3 � ce1 � e2�
�c e2 � be3����ce1 � ae3�
�������������������������������� �����������������������������������
c
The first factor is a vector in the e2 � e3 coordinate bivector, while the second is in the e1 � e3
coordinate bivector.
Graphic of x showing its orthogonality to each of c e2 � be3 , and �c e1 � ae3 .
The complements of each of these vectors will be bivectors which contain the original vector x.
We can verify this easily with GrassmannSimplify.
�������������������
����c e2 � be3���ae1
� be2 � ce3�,
����������������������
��c e1 � ae3���ae1
� be2 � ce3���
�0, 0�
5.10 Complements in a vector subspace of a multiplane
Metrics in a multiplane
If we want to interpret one element of a linear space as an origin point, we need to consider
which forms of metric make sense, or are useful in some degree. We have seen above that a
Euclidean metric makes sense both in vector spaces (as we expected) but also in the plane where
one basis element is interpreted as the origin point and the other two as basis vectors.
More general metrics make sense in vector spaces, because the entities all have the same
interpretation. This leads us to consider hybrid metrics on n-planes in which the vector subspace
has a general metric but the origin is orthogonal to all vectors. We can therefore adopt a
Euclidean metric for the origin, and a more general metric for the vector subspace.
� 1 0 � 0
������������
�������������
0 g11 � g1�n
Gij �
� � � �
� 0 gn1 � gnn �
These hybrid metrics will be useful later when we discuss screw algebra in Chapter 7 and
Mechanics in Chapter 8. Of course the permitted transformations on a space with such a metric
must be restricted to those which maintain the orthogonality of the origin point to the vector
subspace.
2001 4 5
5.33