By Scott Messick
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Extra resources for Limits in category theory
Blades of higher grade can be factored in a similar way. Re versio In algebraic computations, it is often desirable to reorder the factors in a product. For this reason, it is convenient to introduce the operation of reversion defined by the equations (AB)' = B ~ A ~ , ( A + B ) ' = A ' + B', (At4),,= ( A ),,* at = a if a = (a),. We say that A t is the reverse of the multivector A. 24d) that the reverse of a product of vectors is (a3a2.. = a,. . a,a,. This justifies our choice of the name "reverse".
This scalar quantity clearly depends on the orientation of OB, because 08 determines the line from which the angle is measured. By taking orientation into account. we go slightly beyond the Greek idea of perpendicular projection. Define the "inner product" of two directed line segments, denoted by vectors a and b respectively, to be the oriented line segn~entobtained by dilating the projection of a on b by the magnitude of b. b = 1 a 1 1 b cos 8. 1 This expression is conimonly taken as the definition of a'b, but that calls for an independent definition of cos 8, which would be out of place here.
11) reduces to the usual rule for adding areas, Both the inner and outer products are measures of relative direction, but they conzplement one another. Relations which are difficult or impossible to obtain with one rnay be easy to obtain with the other. Whereas the equation a-b = 0 provides a simple expression of "perpendicular", ah b = 0 provides a sirnple expression of "parallel". To illustrate the point, reconsider the vector equation for a triangle, which was analyzed above with the help of the inner product.