Grassmann Algebra

GeometricInterpretations.nb 29
��� ��� � e2��
� � e1 � e2 � e3 ��38 � � 329 e2�
��� ��� � e3��
� � e1 � e2 � e3 ��48 � � 329 e3�
We then take the product of these points (ignoring the weights) to form the plane.
329 e1
329 e2
329 e3
�� ����
� ����������������
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� 13 �
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� ����������������
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� 38 �
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� ����������������
���
� 48 �
To verify that this is indeed the same plane, we can check to see if these points are in the
original plane. For example:
��� � ����
�
�
0
329 e1
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13
Planes in a 4-plane
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�
From the results above, we can expect that a plane in a 4-plane is most economically expressed
as the product of three points, each point lying in one of the coordinate 2-planes. For example:
���� � x1�e1 � x2�e2���� � y2�e2 � y3�e3���� � z3�e3 � z4�e4�
If a plane is expressed in any other form, we can express it in the form above by first
determining its points of intersection with the coordinate planes and then taking the exterior
product of the resulting points. This leads to the following identity:
���� ��� � e1 � e2�� � �� ��� � e2 � e3�� � �� ��� � e3 � e4��
Planes in an m-plane
A plane in an m-plane is most economically expressed as the product of three points, each point
lying in one of the coordinate (mÐ2)-planes.
���� � x1�e1 � � � xi1 �ei1 ������������ � �� � y1�e1 � � � yi3 �ei3 ������������ � �� � z1�e1 � � � zi5 �ei5 ������������ � � � xi2 �ei2 ������������ � � � xm �em �
� � � yi4 �ei4 ������������ � � � ym �em�
� � � zi6 �ei6 ������������ � � � zm �em�
Here the notation xi�ei � means that the term is missing from the sum.
This formulation indicates that a plane in an m-plane has at most 3(mÐ2) independent scalar
parameters required to describe it.
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