Grassmann Algebra

TheInteriorProduct.nb 24
The measure of bound elements
In this section we explore some relationships for measures of bound elements in an n-plane
under the hybrid metric [5.33] introduced in Section 5.10. Essentially this metric has the
properties that:
1. The scalar product of the origin with itself (and hence its measure) is unity.
2. The scalar product of the origin with any vector or multivector is zero.
3. The scalar product of any two vectors is given by the metric tensor on the n-dimensional
vector subspace of the n-plane.
We summarize these properties as follows:
The measure of a point
� ���� � � 1 Α m ���� � � 0 ei ���� ej � gij
Since a point P is defined as the sum of the origin � and the point's position vector Ν relative to
the origin, we can write:
P ���� P � �� �Ν� ���� �� �Ν� � � ���� � � 2�� ���� Ν�Ν���� Ν�1 �Ν���� Ν
�P� 2 � 1 � �Ν� 2
Although this is at first sight a strange result, it is due entirely to the hybrid metric discussed in
Section 5.10. Unlike a vector difference of two points whose measure starts off from zero and
increases continuously as the two points move further apart, the measure of a single point starts
off from unity as it coincides with the origin and increases continuously as it moves further
away from it.
The measure of the difference of two points is, as expected, equal to the measure of the
associated vector.
�P1 � P2� ���� �P1 � P2�
� P1 ���� P1 � 2�P1 ���� P2 � P2 ���� P2
� �1 �Ν1 ���� Ν1� � 2��1 �Ν1 ���� Ν2� � �1 �Ν2 ���� Ν2�
�Ν1���� Ν1 � 2�Ν1 ���� Ν2 �Ν2 ���� Ν2
� �Ν1 �Ν2����� �Ν1 �Ν2�
The measure of a bound multivector
Consider now the more general case of a bound m-vector P� Α with P � �+Ν. Applying
m
formula 6.99 derived in Section 6.10, we can write:
�P � Α m � ���� �P � Α m � � �Α m ���� Α m ���P ���� P� � �Α m ���� P� ���� �Α m ���� P�
But Α m ���� P �Α m ���� �� �Ν� �Α m ���� Ν and P����P � 1+Ν����Ν, so that:
2001 4 5
6.56
6.57