Grassmann Algebra

GeometricInterpretations.nb 6
M1 � 2�P1; P1 � � � e1 � 3�e2 � 4�e3;
M2 � 4�P2; P2 � � � 2�e1 � e2 � 2�e3;
M3 � 7�P3; P3 � � � 5�e1 � 3�e2 � 6�e3;
M4 � 5�P4; P4 � � � 4�e1 � 2�e2 � 9�e3;
We simply add the mass-weighted points.
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M � �
i�1
Mi
5 �� � 4e1 � 2e2 � 9e3� � 7 �� � 5e1 � 3e2 � 6e3� �
2 �� � e1 � 3e2 � 4e3� � 4 �� � 2e1 � e2 � 2e3�
Simplifying this gives a weighted point with weight 18, the scalar attached to the origin.
M � Simplify�M�
18 � � 5e1 � 33 e2 � 103 e3
Taking the weight out as a factor, that is, expressing the result in the form mass � point
M � WeightedPointForm�M�
18 �� � 1
�������
18 ��5 e1 � 33 e2 � 103 e3��
Thus the total mass is 18 situated at the point � � 1
������
18 ��5 e1 � 33 e2 � 103 e3�.
4.3 Geometrically Interpreted 2-Elements
Simple geometrically interpreted 2-elements
It has been seen in Chapter 2 that the linear space � may be generated from � by the exterior
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product operation. In the preceding section the elements of � have been given two geometric
1
interpretations: that of a vector and that of a point. These interpretations in turn generate various
other interpretations for the elements of �. 2
In � there are at first sight three possibilities for simple elements:
2
1. x�y (vector by vector).
2. P�x (point by vector).
3. P1 � P2 (point by point).
However, P1 � P2 may be expressed as P1 ��P2 � P1� which is a point by a vector and thus
reduces to the second case.
There are thus two simple interpreted elements in �: 2
1. x�y (the simple bivector).
2. P�x (the bound vector).
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