$\bf a, b, x$ are n-dimensional vectors. Given that every point $\bf x$ that satisfies $\bf a\cdot x$ $=0$ also satisfies $\bf b\cdot x$$=0$, where we are using the dot product and $\bf b$ is not zero. Prove that $\bf a$$=k\bf b$, where $k$ is a scalar constant.
Background:
I'm trying to convince myself that the Lagrange Multiplier method makes sense (I'm currently limiting my effort to single constraint), and I'm able to boil it down to this lemma. However, I did not phrase my question in the context of Lagrange Multiplier because I think this is just a general lemma that works for other context as well.
I know my question works for $n=2, 3$ because eg when $n=3$, the points $\bf x$ which satisfies $\bf a\cdot x$$=0$ forms a plane with normal $\bf a$. Extending this logic further, we see that $\bf a$ and $\bf b$ are parallel and the result holds. But the idea of perpendicular and parallel breaks down for $n>3$, so I don't know how to proceed for higher dimensions.