Grassmann Algebra

TheComplement.nb 31
The vector z is the position vector of any point on the line defined by the bound vector P
����� . A
particular point of interest is the point on the line closest to the point P or the origin. The
position vector z would then be a vector at right angles to the direction of the line. Thus we can
write e1 � e2 as:
e1 � e2 �� �ae1 � be2���be1 � ae2�
�������������������������������� ��������������������������������
a2 � b2 The final expression for the complement of the point P � � � ae1 � be2 can then be written
as a bound vector in a direction perpendicular to the position vector of P through a point P � .
�����
P �
����
�
�
�ae1 � be2�
������������������������������
a2 � b2 ���
��be1 � ae2� � P
�
� ��be1 � ae2�
Graphic of P, P � , their position vectors, the line joining them, and the line and bound vector
perpendicular to this line through P � .
The point P � is called the inverse point to P. Inverse points are situated on the same line
through the origin, and on opposite sides of it. The product of their distances from the origin is
unity.
� The Euclidean complement in a vector 3-space
Now we come to the classical 3-dimensional vector space which is the usual geometric
interpretation of a 3-dimensional linear space. We start our explorations by generating a palette
of basis elements and their complements.
�3; ComplementPalette
� BASIS COMPLEMENT
�
0
1 e1 � e2 � e3
� l
� l
� l
� 2
� 2
e1
e2
e3
e1 � e2
e1 � e3
e2 � e3
��e1 � e3�
e1 � e2
e3
�e2
�
2
e2 � e3 e1
�
3
e1 � e2 � e3 1
First, we consider a vector x and take its complement.
�����
x � ae1
����� � be2
����� � ce3
����� � ae2 � e3 � be1 � e3 � ce1 � e2
The bivector x
����� is thus a sum of components, each in one of the coordinate bivectors. We have
already shown in Chapter 2: The Exterior Product that a bivector in a 3-space is simple, and
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