Quadratic form of a squared cross product
= Quadratic form of a squared cross product
{title2=$M=|b|^2I-bb^T$}
For real three-dimensional vectors, <Lagrange's identity> gives $|a\times b|^2=a^TMa$ with symmetric matrix $M=|b|^2I-bb^T$. For $b\ne0$, the normal line spanned by $b$ is its <kernel> and the perpendicular plane is its $|b|^2$ <eigenspace>. Consequently $M$ is <positive semidefinite>, has rank two and has zero <determinant>. For $b=0$, it is the zero matrix.