Grassmann Algebra

GeometricInterpretations.nb 21
�P1 � P0���P2 � P0� � �P2 � P1���P0 � P1� � �P0 � P2���P1 � P2�
This property of nilpotence is shared by the boundary operator of algebraic topology and the
exterior derivative. Furthermore, if a product with a given 1-element is considered an operation,
then the exterior, regressive and interior products are all likewise nilpotent.
4.8 Line Coordinates
We have already seen that lines are defined by bound vectors independent of the dimension of
the space. We now look at the types of coordinate descriptions we can use to define lines in
bound spaces (multiplanes) of various dimensions.
For simplicity of exposition we refer to a bound vector as 'a line', rather than as 'defining a line'.
� Lines in a plane
To explore lines in a plane, we first declare the basis of the plane: �2 .
�2
��, e1, e2�
A line in a plane can be written in several forms. The most intuitive form perhaps is as a product
of two points �+x and �+y where x and y are position vectors.
L � �� � x���� � y�
Graphic of a line through two points specified by position vectors.
We can automatically generate a basis form for each of the position vectors x and y by using
the GrassmannAlgebra CreateVector function.
�X � CreateVector�x�, Y� CreateVector�y��
�e1 x1 � e2 x2, e1 y1 � e2 y2�
L � �� � X���� � Y�
L � �� � e1 x1 � e2 x2���� � e1 y1 � e2 y2�
Or, we can express the line as the product of any point in it and a vector parallel to it. For
example:
L � �� � X���Y � X� � �� � Y���Y � X� ��Simplify
L � �� � e1 x1 � e2 x2���e1 ��x1 � y1� � e2 ��x2 � y2�� �
�� � e1 y1 � e2 y2���e1 ��x1 � y1� � e2 ��x2 � y2��
Graphic of a line through a point parallel to the difference between two position vectors.
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