# 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*:

- Distributivity over the sum of vectors:
**x**∧(**y** + **z**) = **x**∧**y** + **x**∧**z**,
- (
*a***x**)∧(*b***y**) = (*ab*)(**x**∧**y**)
- Anticommutativity or antisymmetry:
**x**∧**y** = -**y**∧**x**
- Associativity
- 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.

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 *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.