Grassmann Algebra

TheInteriorProduct.nb 41
� � 2 � � 2
�Α � x� � �Α ���� x� m
m
� 1
Sin�Θ� 2 � � 2
� �Α � x� m
Cos�Θ� 2 � � 2
� �Α ���� x� m
We will explore this more fully in what follows.
� The angle between a vector and a bivector
As a simple example, take the case where x is rewritten as x1 and Α m is interpreted as the
bivector x2 � x3 .
Diagram of three vectors showing the angles between them, and the angle between the
bivector and the vector.
The angle between any two of the vectors is given by formula 6.109.
Cos�Θij� 2 � �xi
� ���� xj
� � 2
6.113
6.114
The cosine of the angle between the vector x1 and the bivector x2 � x3 may be obtained from
formula 6.113.
� � 2
�Α ���� x� m
� ��x2 � x3� ���� x1� ���� ��x2 � x3� ���� x1�
�������������������������������� �������������������������������� �������������
��x2 � x3� ���� �x2 � x3����x1 ���� x1�
To express the right-hand side in terms of angles, let:
A � ��x2 � x3� ���� x1� ���� ��x2 � x3� ���� x1�
�������������������������������� �������������������������������� �������������
��x2 � x3� ���� �x2 � x3����x1 ���� x1� ;
First, convert the interior products to scalar products and then convert the scalar products to
angle form given by formula 6.109. GrassmannAlgebra provides the function ToAngleForm
for doing this in one operation.
ToAngleForm�A�
�Cos�Θ1,2� 2 � Cos�Θ1,3� 2 � 2 Cos�Θ1,2� Cos�Θ1,3� Cos�Θ2,3��
Csc�Θ2,3� 2
This result may be verified by elementary geometry to be Cos�Θ� 2 where Θ is defined on the
diagram above. Thus we see a verification that formula 6.113 permits the calculation of the
angle between a vector and an m-vector in terms of the angles between the given vector and any
m vector factors of the m-vector.
2001 4 5