Suppose we have a fixed point $p \in \mathbb{R}^3$ and a smooth curve $\beta \colon I \to \mathbb{R}^3$ such that $\beta(0)\perp p$ and $\beta'(t) \perp p $ for all $ t \in I$. Is this condition enough to ensure that $\beta(t) \perp p$ for all $t \in I$ ? My work so far is treating the case in which all of the coordinate functions of $\beta$ are analytic. In this case we get that: $$ \langle \beta(t),p\rangle = \sum_{k=0}^\infty \frac{t^k}{k!}\langle \beta^{(k)}(0),p \rangle = 0 $$ But i think that's asking a lot to $\beta$. Any help in the general case would be appreciated.
Note: Smooth curve means having derivatives of all orders.
