= Exterior product
{title2=$\wedge$}
= Wedge product
{synonym}
The <exterior product>, also called the <wedge product>, combines oriented elements through the multiplication
$$
\Lambda^pV\otimes\Lambda^qV\longrightarrow\Lambda^{p+q}V.
$$
For two <vectors>, it is <bilinear> and alternating:
$$
(au+bv)\wedge w=a(u\wedge w)+b(v\wedge w),\qquad
u\wedge(av+bw)=a(u\wedge v)+b(u\wedge w),\qquad
u\wedge u=0.
$$
These rules imply $v\wedge u=-u\wedge v$. They distinguish the <exterior product> from both the symmetric <dot product> and the unconstrained <tensor product>.<Linear Algebra Done Right by Sheldon Axler (4th edition)>[§9B, Definition 9.27 and Theorem 9.30, printed pp. 347–348: alternation and the sign of a swap]{ref}
In an oriented orthonormal plane, write $u=(a,b)$ and $v=(c,d)$. Expanding in its <standard basis> gives
$$
u\wedge v=(ad-bc)e_1\wedge e_2.
$$
The <determinant> $ad-bc$ is the signed area of the <parallelogram> formed by the two inputs. Its <absolute value> is the ordinary area. Reversing the input order reverses the sign; parallel inputs produce zero.
For the original example $u=(3,1)$ and $v=(1,2)$,
$$
u\wedge v=\boxed{5e_1\wedge e_2},\qquad
v\wedge u=-5e_1\wedge e_2.
$$
Replacing $v$ by $v+u=(4,3)$ changes the shape but preserves the product: $3\cdot3-1\cdot4=5$. Both <parallelograms> have the same oriented area element. If $v=2u$, the <exterior product> is zero, since $u\wedge(2u)=2(u\wedge u)=0$.
\Image[/exterior-product-oriented-area-and-shear.png]
{title=An <exterior product> records signed area and survives a shear}
{description=Original <parallelograms> generated by $u=(3,1)$ with $v=(1,2)$ and with $v+u=(4,3)$. Both have signed area $5$ in the chosen Cartesian orientation. Reversing the order of the two input <vectors> gives $-5$. The vertices and arrows are illustrative geometry, rather than measured data.}
The same rule adds directions in higher dimensions. With $w=(2,-1,1)$ and the three-dimensional inputs under <bivector>,
$$
u\wedge v\wedge w
=\det\begin{pmatrix}1&0&2\\2&1&-1\\0&3&1\end{pmatrix}
e_1\wedge e_2\wedge e_3
=16e_1\wedge e_2\wedge e_3.
$$
Its <coefficient> is the <oriented volume> of the three-dimensional <parallelepiped>. Repeating or linearly depending on earlier inputs collapses the volume, giving zero.
For <homogeneous elements of a graded algebra> of degrees $p,q$, the general commutation rule is
$$
\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha.
$$
Swapping a <bivector> with another <bivector> therefore produces a positive sign, since $pq=4$. It is the number of individual vector swaps that controls the sign, not a rule that every pair of objects must anticommute.
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