The properties of the outer product "∧" are, for all vectors x, y, z in V_{n}, and scalars a, b in F:
By virtue of properties (1) and (2), the vector space becomes an algebra, and by property (4) is also associative. The algebra generated is a stepped algebra or graded algebra.
If two vectors x and y are linearly independent (LI), the outer product generates a new entity called bivector. A vector can be seen as a "piece" of a straight line with an orientation; a bivector is a piece of a plane with an orientation. Geometrically a bivector x∧y is the sweeping surface generated when the vector x slips along y in the direction of y. The area of this surface is the magnitude of the bivector, x∧y = x y sin(α), were α is the angle between x and y. The orientation of the bivector is given by spinning from x to y. Thus, reverting the order of the operands reverts the sense or orientation of the bivector, but keeps its magnitude, so behaving exactly as the cross product.
Similarly, the product of a bivector with a third LI vector gives rise to an oriented volume, generated by sliding the bivector "area" along of the third vector. This oriented volume is called trivector. In general, given k LI vectors, their outer product generates a kdimensional volume or kvector.
If we took our vectors from an ndimensional vector space, then we cannot get more than n LI vectors; thus, the outer product of more than n vectors is always 0, and the nvector is the "highest order" kvector that can be generated. Note that this nvector is a representation of the original vector space V_{n}.
The advantages of these new elements are many. A bivector can be used to unambiguously represent a plane embedded in any ndimensional space, while the use of the normal vector is only useful in a 3D space. A kvector thus represents a kdimensional space in any ndimensional space, and this representation does not change when switching to higher dimensional spaces.
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