Rewriting plane as a vector equation

Question

If a plane is given by the vectors $a,x,y$ such as $r=a + m x + n y$ , how can it be rewritten as the vector equation $(r-a).n = 0$?

I understand that $n$ is normal to the plane, but how do I prove this?

Answer 1

Let $\mathbf{n}=\mathbf{x}\times\mathbf{y}$.

Use the fact that $\mathbf{p} \cdot (\mathbf p \times \mathbf q)=\mathbf{q} \cdot (\mathbf p \times \mathbf q)=0$.

$(\mathbf{r}-\mathbf{a})\cdot\mathbf{n}=m\mathbf{x}\cdot(\mathbf{x}\times\mathbf y)+n\mathbf{y}\cdot(\mathbf{x}\times\mathbf y)=0$.