= Bivector
{title2=$w\in\Lambda^2V$}
{wiki}
A <bivector> is an element of the second <exterior power> $\Lambda^2V$. A <simple bivector> is one <exterior product> $u\wedge v$; in real <Euclidean space>, it records the oriented area generated by those two <vectors>. A general <bivector> is a sum of such elements and need not describe just one plane. Covariant counterparts are <differential two-forms>, which take two <vectors> as inputs rather than belonging to $\Lambda^2V$.
In $\mathbb R^3$, choose
$$
u=(1,2,0),\qquad v=(0,1,3).
$$
Expanding by <bilinearity> gives
$$
B=u\wedge v=e_1\wedge e_2+3e_1\wedge e_3+6e_2\wedge e_3.
$$
The three <coefficients> are signed areas of the projections onto the $xy$, $xz$, and $yz$ coordinate planes. They are three components of one <bivector>, not the three coordinates of either input <vector>.
\Image[/bivector-coordinate-projected-areas.png]
{title=One <bivector> has three coordinate-plane area components in three dimensions}
{description=Original <parallelogram> generated by $u=(1,2,0)$ and $v=(0,1,3)$. The three planar panels show its projected signed areas $1$, $3$, and $6$ in the coordinate orders $xy$, $xz$, and $yz$. With the usual <dot product>, the full area is $\sqrt{46}$; projected areas are components, so they are not added to obtain that area.}
With the usual <dot product> and a fixed <orientation of a vector space>, the <cross product> represents this area element by the <normal vector> $(6,-3,1)$. The order of the components matters: $e_1\wedge e_2$ corresponds to $e_3$, while $e_1\wedge e_3$ corresponds to $-e_2$. The <Gram determinant> checks the full area:
$$
\|u\wedge v\|^2
=\|u\|^2\|v\|^2-(u\cdot v)^2
=5\cdot10-2^2=46.
$$
Here the <norm> on <bivectors> is the one induced by the <Euclidean inner product>. The <exterior product> exists without that metric and in arbitrary dimension; the familiar <cross product> description relies on the additional three-dimensional geometry.
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