Lagrange identity for the cross product
= Lagrange identity for the cross product
{c}
= Lagrange's identity
{c}
{synonym}
For three-dimensional vectors,
$$
|x\times y|^2=|x|^2|y|^2-(x\cdot y)^2.
$$
The corresponding identity in $\mathbb R^n$ is
$$
|x|^2|y|^2-(x\cdot y)^2
=\frac12\sum_{i,j}(x_iy_j-x_jy_i)^2.
$$