Grassmann Algebra

TheInteriorProduct.nb 8
ToScalarProducts�Ξ�
Β �x � Α� �Α�x � Β�
We can see from this that Ξ is contained in the bivector Α � Β. We can also verify this by
expanding their exterior product.
ToScalarProducts�Ξ ��Α � Β��
0
The vectors Ξ and x are also orthogonal.
ToScalarProducts����� x�
0
Diagram of a vector x with a component in a bivector, and one orthogonal to it.
6.3 Properties of the Interior Product
Implications of the Complement Axiom
The Complement Axiom was introduced in Chapter 5 in the form Α m � Β k
recast the right-hand side into a form involving the interior product.
���������
Α � Β �
�����
Αm ���� Βk
m k
� ��1� mk � k
����� ���� Αm
��������� ����� �����
� Αm � Βk . We can
����� ��������� ����� ����������� ����� �����
Further, by replacing Α by Α in Αm � Β we have that Αm � Βk � Αm ���� Βk . Putting
m m k
����� ����� m��n�m�
Α � ��1� �Αm yields an alternative definition of the interior product in terms of the
m
exterior product and the complement.
Α m ���� Β k
Extended interior products
� ��1�m��n�m� ����� �����������
�Α � Βk
m
The interior product of an element with the exterior product of several elements may be shown
straightforwardly to be a type of 'extended' interior product.
2001 4 5
6.19
6.20