= Plane through a point and a line given by a cross product
{title2=$\mathbf r\cdot(\mathbf b+\mathbf a\times\mathbf y)=\mathbf y\cdot\mathbf b$}
Suppose $\mathbf a\ne\mathbf0$ and $\mathbf a\cdot\mathbf b=0$. The <line> $\mathbf r\times\mathbf a=\mathbf b$ has <parametrization> $\mathbf r=\mathbf r_0+t\mathbf a$, where $\mathbf r_0=(\mathbf a\times\mathbf b)/|\mathbf a|^2$. For a <point> $\mathbf y$ outside the <line>, a <normal vector> of the unique <plane> containing both is $\mathbf N=\mathbf a\times(\mathbf y-\mathbf r_0)=\mathbf b+\mathbf a\times\mathbf y$. Thus its equation is the one in the title. The <vector triple product> verifies $\mathbf r_0\times\mathbf a=\mathbf b$, and cyclic invariance of the <scalar triple product> verifies that every <point> of the <line> satisfies the equation. The excluded case $\mathbf N=\mathbf0$ is exactly $\mathbf y$ lying on the <line>, when the <plane> is not unique.
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