Grassmann Algebra

GeometricInterpretations.nb 7
A note on terminology
The term bound as in the bound vector P�x indicates that the vector x is conceived of as bound
through the point P, rather than to the point P, since the latter conception would give the
incorrect impression that the vector was located at the point P. By adhering to the terminology
bound through, we get a slightly more correct impression of the 'freedom' that the vector enjoys.
The term 'bound vector' is often used in engineering to specify the type of quantity a force is.
The bivector
Earlier in this chapter, a vector x was depicted graphically by a directed and sensed line
segment supposed to be located nowhere in particular.
Graphic showing an arrow with symbol x attached.
In like manner a simple bivector may be depicted graphically by an oriented plane segment also
supposed located nowhere in particular.
Graphic showing a parallelogram constructed from two vectors x and y with common tails.
The symbol x�y is attached to the parallelogram.
Orientation is a relative concept. The plane segment depicting the bivector y�x is of opposite
orientation to that depicting x�y.
Graphic showing a parallelogram y�x .
The oriented plane segment or parallelogram depiction of the simple bivector is misleading in
two main respects. It incorrectly suggests a specific location in the plane and shape of the
parallelogram. Indeed, since x�y � x�(x+y), another valid depiction of this simple bivector
would be a parallelogram with sides constructed from vectors x and x+y.
Graphic showing parallelograms x�y and x�(x+y) superimposed.
In the following chapter a metric will be introduced onto � from which a metric is induced onto
1
�. This will permit the definition of the measure of a vector (its length) and the measure of the
2
simple bivector (its area).
The measure of a simple bivector is geometrically interpreted as the area of the parallelogram
formed by any two vectors in terms of which the simple bivector can be expressed. For
example, the area of the parallelograms in the previous two figures are the same. From this point
of view the parallelogram depiction is correctly suggestive, although the parallelogram is not of
fixed shape. However, a bivector is as independent of the vector factors used to express it as any
area is of its shape. Strictly speaking therefore, a bivector may be interpreted as a portion of an
(unlocated) plane of any shape. In a metric space, this portion will have an area.
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