Grassmann Algebra

TheInteriorProduct.nb 37
The measure of a cross product
The measure of the cross product of two elements is equal to the measure of their exterior
product.
�Α �Β� � �Α � Β� m k m k
6.10 The Triangle Formulae
Triangle components
6.98
In this section several formulae will be developed which are generalizations of the fundamental
Pythagorean relationship for the right-angled triangle and the trigonometric identity
cos 2 �Θ� � sin 2 �Θ� � 1. These formulae will be found useful in determining projections, shortest
distances between multiplanes, and a variety of other geometric results. The starting point is the
product formula [6.66] with k = m.
�Α ���� Β��x � �Α � x� ���� Β � ��1�
m m m m
m�1 �Α ���� �Β ���� x�
m m
By dividing through by the inner product Α ���� Β, we obtain a decomposition of the 1-element x
m m
into two components.
Now take formula 6.71:
x �
�Α m � x� ���� Β m
������������������������
Α m ���� Β m
� ��1� m�1 �
Α ���� �Β ���� x�
m m
������������������������
Α ���� Β
m m
��Α ���� Β��x1� ���� x2 � �Α � x1� ���� �Β � x2� � ��1�
m k
m k
m�k �� Α ���� x2� ���� �Β ���� x1�
m k
and put k = m, x1 equal to x and x2 equal to z.
�Α ���� Β���x ���� z� � �Α � x� ���� �Β � z� � �Α ���� z� ���� �Β ���� x�
m m
m m
m m
When Α is equal to Β and x is equal to z, this reduces to a relationship similar to the
m m
fundamental trigonometric identity.
2001 4 5
� � � �
x ���� x � �Α � x� ���� �Αm � x� � �Αm ���� x� ���� �Αm ���� x�
m
6.99
6.100
6.101