= Equation of a plane through three points
{title2=$(\mathbf r-\mathbf a)\cdot[(\mathbf b-\mathbf a)\times(\mathbf c-\mathbf a)]=0$}
For three non-collinear points with position <vectors> $\mathbf a,\mathbf b,\mathbf c$, a nonzero <normal vector> is
$$
\mathbf N=(\mathbf b-\mathbf a)\times(\mathbf c-\mathbf a).
$$
The <plane> is the set of $\mathbf r$ satisfying $(\mathbf r-\mathbf a)\cdot\mathbf N=0$. Expanding the <cross product> and using the <scalar triple product> gives
$$
\mathbf r\cdot(\mathbf a\times\mathbf b+\mathbf b\times\mathbf c+\mathbf c\times\mathbf a)
=\mathbf a\cdot(\mathbf b\times\mathbf c).
$$
Non-collinearity is essential: otherwise $\mathbf N=0$ and the equation fails to select a unique plane.
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