Grassmann Algebra

TheInteriorProduct.nb 43
Note that this has been simplified somewhat as permitted by the symmetry of the scalar product.
By putting this in its angle form we get the usual expression for the volume of a parallelepiped:
ToAngleForm�V�
�
���Α1� 2 �Α2�2 �Α3�2 ��1 � Cos�Θ1,2�2 � Cos�Θ1,3�2 �
2 Cos�Θ1,2� Cos�Θ1,3� Cos�Θ2,3� � Cos�Θ2,3� 2 ��
A slight rearrangement gives the volume of the parallelepiped as:
V � �Α1� �Α2� �Α3��
�
�1 � 2 Cos�Θ1,2� Cos�Θ1,3� Cos�Θ2,3� � Cos�Θ1,2� 2 �
Cos�Θ1,3� 2 � Cos�Θ2,3� 2 �
We can of course use the same approach in any number of dimensions. For example, the
'volume' of a 4-dimensional parallelepiped in terms of the lengths of its sides and the angles
between them is:
ToAngleForm�Measure�Α1 � Α2 � Α3 � Α4��
�
��Α1�2 �Α2�2 �Α3�2 �Α4�2 �1 � Cos�Θ2,3�2 � Cos�Θ2,4�2 �
2 Cos�Θ1,3� Cos�Θ1,4� Cos�Θ3,4� � Cos�Θ3,4�2 �
2 Cos�Θ2,3� Cos�Θ2,4���Cos�Θ1,3� Cos�Θ1,4� � Cos�Θ3,4�� �
2 Cos�Θ1,2� �Cos�Θ1,4� �Cos�Θ2,4� � Cos�Θ2,3� Cos�Θ3,4�� �
Cos�Θ1,3� �Cos�Θ2,3� � Cos�Θ2,4� Cos�Θ3,4��� �
Cos�Θ1,4�2 Sin�Θ2,3� 2 � Cos�Θ1,3�2 Sin�Θ2,4� 2 �
Cos�Θ1,2�2 Sin�Θ3,4� 2 ��
6.12 Projection
To be completed.
6.13 Interior Products of Interpreted Elements
To be completed.
6.14 The Closest Approach of Multiplanes
To be completed.
2001 4 5
6.116