Outer product

The outer product or wedge product is a non-closed vector product defined in a vector space V over a scalar field F. It can be seen as a generalization to n dimensions of the Gibbs vectorial product or cross product, which can only be defined in vector spaces of 3 or 7 dimensions.

The properties of the outer product "∧" are, for all vectors x, y, z in V_n, and scalars a, b in F:

1. Distributivity over the sum of vectors: x∧(y + z) = x∧y + x∧z
2. (ax)∧(by) = (ab)(x∧y)
3. Anticommutativity or antisymmetry: x∧y = −y∧x
4. Associativity: (x∧y)∧z = x∧(y∧z)
5. If x and y are linearly dependent, then x∧y = 0

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.

k-vectors

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), where 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 k-dimensional volume or k-vector.

If we took our vectors from an n-dimensional 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 n-vector is the "highest order" k-vector that can be generated. Note that this n-vector 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 n-dimensional space, while the use of the normal vector is only useful in a 3D space. A k-vector thus represents a k-dimensional space in any n-dimensional space, and this representation does not change when switching to higher dimensional spaces.