I have this Algebra question:
Prove that the equation for the intersecting line between the planes $\vec{r}\cdot\vec{n_1}=0$ and $\vec{r}\cdot\vec{n_2}=0$ is given by $\vec{r}=t(\vec{n_1}\times\vec{n_2})$.
I can see from the equations that both planes go through origin and that a cross product vector of the normals will give the line vector through the origin, which is what the question is about. The problem is how to prove this in a correct way.
Is it enough to replace $\vec{r}$ in the first two equations which are still true: $$t(\vec{n_1}\times\vec{n_2})\cdot\vec{n_1}=0$$$$t(\vec{n_1}\times\vec{n_2})\cdot\vec{n_1}=0$$
The line vector lies in both planes and since both planes go through origin the intersecting line will as well.
How do I do this with some beautiful math jargon that will satisfy the grumpiest of math professors?
Than you.