Show every point of a line lies on a plane

As in the preceding example, use $\mathbf m=(2,1,4)$ and $\mathbf n=(1,2,-1)$:

$\mathbf m\cdot\mathbf n=2+2-4=0.$

The direction is parallel to the plane. This time the base point satisfies the plane equation:

$\begin{aligned}&\mathbf a\cdot\mathbf n\\ & =(1,1,0)\cdot(1,2,-1)\\ & =3.\end{aligned}$

For every real $\lambda$, $(\mathbf a+\lambda\mathbf m)\cdot\mathbf n=3+\lambda(0)=3$. Therefore all points of the line are on the plane. The source’s preliminary statement “line is parallel to plane” concerns its direction; the final conclusion is containment, with infinitely many common points. Changing the plane constant from 4 in Example 32 to 3 is essential.