Grassmann Algebra

TheInteriorProduct.nb 25
�P � Α m � ���� �P � Α m � � �Α m ���� Α m ���1 �Ν���� Ν� � �Α m ���� Ν� ���� �Α m ���� Ν�
By applying formula 6.99 again this may be re-expressed as:
�P � Α m � ���� �P � Α m � �Α m ���� Α m � �Α m � Ν� ���� �Α m � Ν�
�P � Α� m 2
� �Α� m 2
� �Α � Ν�
m 2
� 2
�P � Α� m
� 2
� 1 � �Α � Ν�
m
Note that the measure of a multivector bound through the origin (Ν � 0) is just the measure of
the multivector.
�� � Α m � � �Α m �
�
�� � Α� � 1
m
If Ν ���� is the component of Ν orthogonal to Α m , then by formula 6.99 we have:
�Α m � Ν� ���� �Α m � Ν� � �Α m ���� Α m ���Ν ���� ���� Ν ���� �
We can thus write formulae 6.58 and 6.59 in the alternative forms:
�P � Α m � 2
� �Α m � 2
��1 � �Ν ���� � 2 �
� 2
�P � Α� � 1 � �Ν
m
���� � 2
Thus the measure of a bound unit multivector indicates its minimum distance from the origin.
2001 4 5
6.58
6.59
6.60
6.61
6.62
6.63
6.64