Grassmann Algebra

TheInteriorProduct.nb 40
� ����� �
Since the proof of formula 6.103 is valid for any simple Α it is also valid for Αm (which is equal
m
to
�����
Α
m
� ) since this is also simple. The proof of formula 6.104 is then completed by applying
formula 6.107.
The next three sections will look at some applications of these formulae.
6.11 Angle
Defining the angle between elements
The angle between 1-elements
The interior product enables us to define, as is standard practice, the angle between two 1elements
Cos�Θ� 2 � �x1
� ���� x2
� � 2
Diagram of two vectors showing the angle between them.
However, putting Α m equal to x1 and x equal to x2 in formula 6.101 yields:
�x1
� � x2
� � 2 � �x1
� ���� x2
� � 2 � 1
This, together with the definition for angle above implies that:
Sin�Θ� 2 � �x1
� � x2
� � 2
This is a formula equivalent to the three-dimensional cross product formulation:
Sin�Θ� 2 � �x1
� � x2
� � 2
but one which is not restricted to three-dimensional space.
Thus formula 6.110 is an identity equivalent to sin 2 �Θ� � cos 2 �Θ� � 1.
The angle between a 1-element and a simple m-element
6.110
6.111
6.112
One may however carry this concept further, and show that it is meaningful to talk about the
angle between a 1-element x and a simple m-element Α for which the general formula [6.102]
m
holds.
2001 4 5