The identity $(u\times v)\cdot(x\times y)=\begin{vmatrix}u\cdot x&v\cdot x\\ u\cdot y&v\cdot y\\\end{vmatrix}$ I know by brutal calculation this identity holds always:
$$(u × v) \cdot (x × y) = \begin{vmatrix} 
u \cdot x & v \cdot x \\ 
u \cdot y & v \cdot y \\
 \end{vmatrix}$$
for arbitrary vectors $u$, $v$, $x$, $y$.
I'd like to know where it come from naturally.
Thank you.
 A: I suppose you are talking about dot products and cross products in $\mathbb R^3$. The identity can be seen as a special case of Cauchy-Binet formula. Alternatively, note that
$$
\pmatrix{x^T\\ y^T\\ (u\times v)^T}\pmatrix{u&v&x\times y}
=\pmatrix{u\cdot x&v\cdot x&0\\ u\cdot y&v\cdot y&0\\ 0&0&(u\times v)\cdot(x\times y)}.\tag{1}
$$
Therefore the RHS has determinant
$$
\det\pmatrix{u\cdot x&v\cdot x\\ u\cdot y&v\cdot y}[(u\times v)\cdot(x\times y)].\tag{2}
$$
On the other hand, since $\det\pmatrix{p&q&r}\equiv(p\times q)\cdot r$ (this actually is the definition of cross product), the LHS of $(1)$ has determinant
$$
[(u\times v)\cdot(x\times y)]^2.\tag{3}
$$
The result now follows by equating $(2)$ and $(3)$ with the use of a continuity argument.
A: Since $\epsilon_{ijk}\epsilon_{ilm}=\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}$, the dot product is$$\epsilon_{ijk}u_jv_k\epsilon_{ilm}x_ly_m=(\delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl})u_jv_kx_ly_m=(u\cdot x)(v\cdot y)-(u\cdot y)(v\cdot x)=\begin{vmatrix}
u\cdot x & v\cdot x \\ 
u\cdot y & v\cdot y \\
\end{vmatrix}.$$This is also the only linear combination of the four vectors, with coefficient $1$ for $u_1v_2x_1y_2$, that's symmetric under simultaneously exchanging $u,\,x$ and $v,\,y$, but antisymmetric under exchanging $u$ with $v$, or $x$ with $y$.
