Grassmann Algebra

TheInteriorProduct.nb 34
Implications of the axioms for the cross product
By expressing one or more elements as a complement, axioms for the exterior and regressive
products may be rewritten in terms of the cross product, thus yielding some of its more
fundamental properties.
� � 6: The cross product of an m-element and a k-element is an (nÐ(m+k))-element.
The grade of the cross product of an m-element and a k-element is nÐ(m+k).
Α m �� m , Β k
�� k � Α m �Β k
� � 7: The cross product is not associative.
� �
n��m�k�
The cross product is not associative. However, it can be expressed in terms of exterior and
interior products.
�Α m �Β k
Α m � �Β k
�Γ r
� �Γ r
� ��1� �n�1���m�k� �Α m � Β k
� � ��1� �n��k�r����m�k�r� � k
� � 8: The cross product with unity yields the complement.
� ���� à r
� Γ � ���� Α
r m
The cross product of an element with the unit scalar 1 yields its complement. Thus the
complement operation may be viewed as the cross product with unity.
1 �Α � Α�1 �
�����
Α
m m m
The cross product of an element with a scalar yields that scalar multiple of its complement.
a �Α � Α�a � a
�����
Α
m m m
� � 9: The cross product of unity with itself is the unit n-element.
The cross product of 1 with itself is the unit n-element. The cross product of a scalar and its
reciprocal is the unit n-element.
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