Grassmann Algebra

GeometricInterpretations.nb 9
The geometric interpretation of the addition of such a simple bivector to a bound vector is then
similar to that for the addition of a vector to a point, that is, a shift in position.
� Sums of bound vectors
A sum of bound vectors � Pi � xi (except in the case � xi � 0) may always be reduced to the
sum of a bound vector and a bivector, since, by choosing an arbitrary point P, � Pi � xi may
always be written in the form:
� Pi � xi � P ��� xi� ����Pi � P��xi
If � xi � 0 then the sum is a bivector. This transformation is of fundamental importance in our
exploration of mechanics in Chapter 8.
Example: Reducing a sum of bound vectors
In this example we verify that a sum of any number of bound vectors can always be reduced to
the sum of a bound vector and a bivector. Note however that it is only in two or three vector
dimensions that the bivector is necessarily simple. We begin by declaring the bound vector
space of three vector dimensions, �3 .
�3
��, e1, e2, e3�
Next, we define, enter, then sum four bound vectors.
Β1 � P1 � x1; P1 � � � e1 � 3�e2 � 4�e3; x1 � e1 � e3;
Β2 � P2 � x2; P2 � � � 2�e1 � e2 � 2�e3; x2 � e1 � e2 � e3;
Β3 � P3 � x3; P3 � � � 5�e1 � 3�e2 � 6�e3; x3 � 2�e1 � 3�e2;
Β4 � P4 � x4; P4 � � � 4�e1 � 2�e2 � 9�e3; x4 � 5�e3;
4
B � �
i�1
Βi
4.1
�� � 4e1 � 2e2 � 9e3���5e3� � �� � 5e1 � 3e2 � 6e3���2e1 � 3e2� �
�� � e1 � 3e2 � 4e3���e1 � e3� � �� � 2e1 � e2 � 2e3���e1 � e2 � e3�
By expanding these products, simplifying and collecting terms, we obtain the sum of a bound
vector (through the origin) � ��4e1 � 2e2 � 5e3� and a bivector
�25 e1 � e2 � 39 e1 � e3 � 22 e2 � e3 . We can use GrassmannSimplify to do the
computations for us.
��B�
� ��4e1 � 2e2 � 5e3� � 25 e1 � e2 � 39 e1 � e3 � 22 e2 � e3
We could just as well have expressed this 2-element as bound through (for example) the point
� � e1 . To do this, we simply add e1 ��4e1 � 2e2 � 5e3� to the bound vector and
subtract it from the bivector to get:
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