Grassmann Algebra

TheInteriorProduct.nb 33
Cross products involving 1-elements
For 1-elements xi the definition above has the following consequences, independent of the
dimension of the space.
The triple cross product
The triple cross product is a 1-element in any number of dimensions.
�������������������������
�x1 � x2� � x3 �
�������������
�x1 � x2��x3
� �x1 � x2� ���� x3
�x1 � x2� � x3 � �x1 � x2� ���� x3 � �x3 ���� x1��x2 � �x3 ���� x2��x1
The box product or triple vector product
The box product, or triple vector product, is an (nÐ3)-element, and therefore a scalar only in
three dimensions.
�x1 � x2� ���� x3 �
�������������
�x1 � x2�
���� x3 �
��������������������
x1 � x2 � x3
�x1 � x2� ���� x3 �
��������������������
x1 � x2 � x3
6.77
6.78
Because Grassmann identified n-elements with their scalar Euclidean complements (see the
historical note in the introduction to Chapter 5), he considered x1 � x2 � x3 in a 3-space to be a
scalar. His notation for the exterior product of elements was to use square brackets
�x1�x2�x3�, thus originating the 'box' product notation for �x1 � x2� ���� x3 used in the threedimensional
vector calculus in the early decades of the twentieth century.
The scalar product of two cross products
The scalar product of two cross products is a scalar in any number of dimensions.
�x1 � x2� ���� �x3 � x4� �
�������������
�x1 � x2�
����
�������������
�x3 � x4�
� �x1 � x2� ���� �x3 � x4�
�x1 � x2� ���� �x3 � x4� � �x1 � x2� ���� �x3 � x4�
The cross product of two cross products
The cross product of two cross products is a (4Ðn)-element, and therefore a 1-element only in
three dimensions. It corresponds to the regressive product of two exterior products.
�������������������������������������
�x1 � x2� � �x3 � x4� �
�������������
�x1 �
�������������
x2���x3
� x4�
� �x1 � x2���x3 � x4�
2001 4 5
�x1 � x2� � �x3 � x4� � �x1 � x2���x3 � x4�
6.79
6.80