Grassmann Algebra

GeometricInterpretations.nb 33
P1 � � � u1 e1 � u1 2 �e2 � u1 3 �e3;
P2 � � � u2 e1 � u2 2 �e2 � u2 3 �e3;
P3 � � � u3 e1 � u3 2 �e2 � u3 3 �e3;
The osculating planes at these three points are:
�1 � � � e1 � e2 � 3u1 � � e1 � e3 � 3u1 2 � � e2 � e3 � u1 3 e1 � e2 � e3;
�2 � � � e1 � e2 � 3u2 � � e1 � e3 � 3u2 2 � � e2 � e3 � u2 3 e1 � e2 � e3;
�3 � � � e1 � e2 � 3u3 � � e1 � e3 � 3u3 2 � � e2 � e3 � u3 3 e1 � e2 � e3;
The point of intersection of these three planes may be obtained by calculating their regressive
product.
���1 � �2 � �3�
� � e1 � e2 � e3 � � � e1 � e2 � e3 ���9 � �u1 � u2� �u1 � u3� �u2 � u3� �
3e1 �u1 � u2� �u1 � u3� ��u1 � u2 � u3��u2 � u3� �
9e3 u1 �u1 � u2� u2 �u1 � u3� �u2 � u3� u3 �
3e2 �u1 � u2� �u1 � u3� �u2 � u3� ��u2 u3 � u1 �u2 � u3���
This expression is congruent to the point of intersection which we write more simply as:
Q �
� � 1 ���� ��u1 � u2 � u3��e1 �
3 1
���� ��u1 u2 � u2 u3 � u3 u1��e2 � �u1 u2 u3��e3;
3
Finally, to show that this point of intersection Q is coplanar with the points P1 , P2 , and P3 , we
compute their exterior product.
��P1 � P2 � P3 � Q�
0
This proves the original assertion.
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