solution curves of $\frac{dx}{u} = \frac{dy}{v} =\frac{dz}{w}$ cut the surfaces $u dx + v dy +w dz =0$ orthogonally?

Question

How can we show that the solution curves of $\frac{dx}{u} = \frac{dy}{v} =\frac{dz}{w}$ cut the surfaces $u dx + v dy +w dz =0$ orthogonally?

I could only think of expressing $u dx + v dy +w dz =0$ as $([u,v,w].[dx,dy,dz] = 0)$, it is intuitive now that the vector $[u,v,w]$ is orthogonal to $[dx,dy,dz]$.

But how to prove it?