Probability of random vector lying on a hyperplane

Probability of random vector lying on a hyperplane

I have a random vector $v \in \mathbb R^n$, of which the elements are
independent. Now there is also a hyperplane $S \subseteq \mathbb R^n$ of
dimension $n-1$. The vector is drawn from any continuous probability
distribution. Now my common sense tells me that the probability that the
vector lies on the hyperplane, is zero ($P(v\in S)=0$). But how would I
prove this? And is this even true?