Grassmann Algebra

TheInteriorProduct.nb 2
6.9 The Cross Product
Defining a generalized cross product
Cross products involving 1-elements
Implications of the axioms for the cross product
The cross product as a universal product
Cross product formulae
6.10 The Triangle Formulae
Triangle components
The measure of the triangle components
Equivalent forms for the triangle components
6.11 Angle
Defining the angle between elements
� The angle between a vector and a bivector
� The angle between two bivectors
� The volume of a parallelepiped
6.12 Projection
To be completed.
6.13 Interior Products of Interpreted Elements
To be completed.
6.14 The Closest Approach of Multiplanes
To be completed.
6.1 Introduction
To this point we have defined three important operations associated with a linear space: the
exterior product, the regressive product, and the complement.
In this chapter we introduce the interior product, the fourth operation of fundamental
importance to Grassmann algebra. The interior product of two elements is defined as the
regressive product of one element with the complement of the other. Whilst the exterior product
of an m-element and a k-element generates an (m+k)-element, the interior product of an melement
and a k-element (m ³ k) generates an (mÐk)-element. This means that the interior
product of two elements of equal grade is a 0-element (or scalar). The interior product of an
element with itself is a scalar, and it is this scalar that is used to define the measure of the
element.
The interior product of two 1-elements corresponds to the usual notion of inner, scalar, or dot
product. But we will see that the notion of measure is not restricted to 1-elements. Just as one
may associate the measure of a vector with a length, the measure of a bivector may be
associated with an area, and the measure of a trivector with a volume.
If the exterior product of an m-element and a 1-element is zero, then it is known that the 1element
is contained in the m-element. If the interior product of an m-element and a 1-element is
zero, then this means that the 1-element is contained in the complement of the m-element. In this
case it may be said that the 1-element is orthogonal to the m-element.
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