By Dorst L., Fontijne D., Mann S.
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If you have a coordinate system, there may be an advantage to redrawing the bivector to have sides to be aligned with the coordinate vectors. But you should realize that there is no unique way of doing this; since a ∧ b = (2a) ∧ (b/2), and so on, the magnitudes of the components are adjustable (as long as area and orientation remain the same). 3: Bivector representations. In fact, as soon as we have computed a bivector quantity, we have lost the identity of the vectors that generated it. 3(f) (as long as we realize that even this circular shape is arbitrary).
In linear algebra, the orientation and the area measure are both well represented by the determinant of a matrix made of the two spanning vectors a and b of the plane: the orientation is its sign, the area measure its weight (both relative to orientation and area measure of the basis used to specify the coordinates of a and b). In 2-D, this specifies an area element of the plane. In 3-D, such an area element would be incomplete without a specification of the attitude of the plane in which it resides.
The outer product, which is the product of spanning and weighting, does not need one. Yet the lengths, areas, and volumes that can be computed using the outer product appear to have a metric feeling to them. 1: Algebraic definition of the terms we use to denote the geometrical properties of a subspace as represented by a blade A. Term Definition Attitude The equivalence class λA, for any λ ∈ R (Relative) weight The value of λ in A = λ I (where I is a selected standard subspace with the same attitude) (Relative) orientation The sign of the weight relative to I SPANNING ORIENTED SUBSPACES 44 the same plane through the origin, and that volumes similarly are ratios of volumes in the same space.
Geometric Algebra for Computer Science by Dorst L., Fontijne D., Mann S.