Grassmann Algebra

GeometricInterpretations.nb 22
Alternatively, we can express L without specific reference to points in it. For example:
L � � ��ae1 � be2� � ce1 � e2
The first term � ��ae1 � be2� is a vector bound through the origin, and hence defines a line
through the origin. The second term c e1 � e2 is a bivector whose addition represents a shift in
the line parallel to itself, away from the origin.
Graphic showing a line through the origin plus a shift due to the addition of a bivector. Don't
draw the basis vectors at right angles.
We know that this can indeed represent a line since we can factorize it into any of the forms:
¥ A line of gradient b
����
through the point with coordinate
a c
���� on the � � e1 axis.
b
L � �� � c
����
b �e1���ae1 � be2�
Graphic of a line through a point on the axis with given gradient.
¥ A line of gradient b
����
through the point with coordinate �
a c
���� on the � � e2 axis.
a
L � �� � c
����
a �e2���ae1 � be2�
Graphic of a line through a point on the axis with given gradient.
¥ Or, a line through both points.
L � ab
�������� ��� �
c c
���� �e2���� �
a c
���� �e1�
b
Of course the scalar factor ab
������
c is inessential so we can just as well say:
L � �� � c
���� �e2���� �
a c
���� �e1�
b
Graphic of a line through a point on each axis.
� Information required to express a line in a plane
The expression of the line above in terms of a pair of points requires the four coordinates of the
points. Expressed without specific reference to points, we seem to need three parameters.
However, the last expression shows, as expected, that it is really only two parameters that are
necessary (viz y = m x + c).
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