Grassmann Algebra

TheInteriorProduct.nb 32
�Α m � Β k
x p � x1
r
Interior Product Formula 7
p
� ���� x � � ���1�
p r�0
rm Ν
���
i�1
�
�
� x1 � x2 � x2 � � � xΝ
p�r r p�r
r
���Α ���� xi
m p�r
����
��Β
� k
���� xi�
r
� xΝ , Ν��
p�r
p
r �
In formula 6.72 putting Β equal to Α expresses a simple p-element in terms of products with a
k
m
unit m-element.
x p � x1
r
p
x � � ���1�
p r�0
r��n�m� Ν
���
i�1
� ���Α
� �
� xi
���
�
���� �Α ���� xi�
�m
p�r�
m r
� x1 � x2 � x2 � � � xΝ
p�r r p�r
r
6.9 The Cross Product
Defining a generalized cross product
� xΝ , Ν��
p�r
p
r �
The cross or vector product of the three-dimensional vector calculus of Gibbs et al. [Gibbs
1928] corresponds to two operations in Grassmann's more general calculus. Taking the crossproduct
of two vectors in three dimensions corresponds to taking the complement of their
exterior product. However, whilst the usual cross product formulation is valid only for vectors
in three dimensions, the exterior product formulation is valid for elements of any grade in any
number of dimensions. Therefore the opportunity exists to generalize the concept.
Because our generalization reduces to the usual definition under the usual circumstances, we
take the liberty of continuing to refer to the generalized cross product as, simply, the cross
product.
The cross product of Α m and Β k
is denoted Α �Β and is defined as the complement of their
m k
exterior product. The cross product of an m-element and a k-element is thus an (nÐ(m+k))element.
Α m �Β k
���������
�Α�Β m k
6.74
This definition preserves the basic property of the cross product: that the cross product of two
elements is an element orthogonal to both, and reduces to the usual notion for vectors in a three
dimensional metric vector space.
2001 4 5
6.75
6.76