Grassmann Algebra

GeometricInterpretations.nb 28
��L ��� � x�
Graphic of a plane with the preceding definition.
Or, we can express it as the product of any line in it and any vector parallel to it (but not parallel
to the line). For example:
��L ��z � x�
Graphic of a plane with the preceding definition.
Given a basis, we can always express the plane in terms of the coordinates of the points or
vectors in the expressions above. However the form which requires the least number of
coordinates is that which expresses the plane as the exterior product of its three points of
intersection with the coordinate axes.
���� � ae1���� � be2���� � ce3�
Graphic of a plane with the preceding definition.
If the plane is parallel to one of the coordinate axes, say � � e 3 , it may be expressed as:
���� � ae1���� � be2��e3
Whereas, if it is parallel to two of the coordinate axes, say � � e 2 and � � e 3 , it may be
expressed as:
���� � ae1��e2 � e3
If we wish to express a plane as the exterior product of its intersection points with the
coordinate axes, we first determine its points of intersection with the axes and then take the
exterior product of the resulting points. This leads to the following identity:
���� ��� � e1����� ��� � e2�� � �� ��� � e3��
Example: To express a plane in terms of its intersections with the coordinate
axes
Suppose we have a plane in a 3-plane defined by three points.
�3; ���� � e1 � 2�e2 � 5�e3���� � e1 � 9�e2���� � 7�e1 � 6�e2 � 4�e3�
�� � e1 � 2e2 � 5e3���� � e1 � 9e2���� � 7e1 � 6e2 � 4e3�
To express this plane in terms of its intersections with the coordinate axes we calculate the
intersection points with the axes.
2001 4 5
��� ��� � e1��
� � e1 � e2 � e3 ��13 � � 329 e1�