Grassmann Algebra

GeometricInterpretations.nb 27
Alternatively, a line in an m-plane � � e1 � e2 � � � em can be expressed in terms of its
intersections with two of its coordinate (mÐ1)-planes, � � e1 � � � ���� i � � � em and
� � e1 � � � ���� j � � � em say. The notation ���� i means that the ith element or term is missing.
L �
�� � x1�e1 � � � ���� i � � � xm �em ���� � y1�e1 � � � ���� j � � � ym �em �
This formulation indicates that a line in m-space has at most 2(mÐ1) independent parameters
required to describe it.
4.12
It also implies that in the special case when the line lies in one of the coordinate (mÐ1)-spaces, it
can be even more economically expressed as the product of two points, each lying in one of the
coordinate (mÐ2)-spaces contained in the (mÐ1)-space. And so on.
4.9 Plane Coordinates
We have already seen that planes are defined by simple bound bivectors independent of the
dimension of the space. We now look at the types of coordinate descriptions we can use to
define planes in bound spaces (multiplanes) of various dimensions.
� Planes in a 3-plane
A plane � in a 3-plane can be written in several forms. The most intuitive form perhaps is as a
product of three non-collinear points �+x, �+y and �+z, where x, y and z are vectors.
���� � x���� � y���� � z�
Graphic of a plane with the preceding definition.
Or, we can express it as the product of any two different points in it and a vector parallel to it
(but not in the direction of the line joining the two points). For example:
���� � x���� � y���z � x�
Graphic of a plane with the preceding definition.
Or, we can express it as the product of any point in it and any two independent vectors parallel
to it. For example:
���� � x���y � x���z � x�
Graphic of a plane with the preceding definition.
Or, we can express it as the product of any line in it and any point in it not in the line. For
example:
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