Grassmann Algebra

TheInteriorProduct.nb 35
1 � 1 � a � 1
���� � 1
a �����
� � 10: The cross product of two 1-elements anti-commutes.
The cross product of two elements is equal (apart from a possible sign) to their cross product in
reverse order. The cross product of two 1-elements is anti-commutative, just as is the exterior
product.
Α m �Β k
� � 11: The cross product with zero is zero.
� ��1� mk  k
�Α m
0 �Α m � 0 � Α m � 0
� � 12: The cross product is both left and right distributive under addition.
�Α m �Β m
Α m � �Β r
� �Γ r
� Α m �Γ r
��à m r
�Γ� � Α�Β�Α�Γ r m r m r
The cross product as a universal product
We have already shown that all products can be expressed in terms of the exterior product and
the complement operation. Additionally, we have shown above that the complement operation
may be written as the cross product with unity.
�����
Α � 1 �Αm � Α�1 m
m
We can therefore write the exterior, regressive, and interior products in terms only of the cross
product.
2001 4 5
6.86
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6.91