I understand the usual definition of a tangent vector in terms of differential operators.
However, suppose I have three points on a Riemannian manifold, one with coordinates $\theta$, one with coordinates $\theta+\varepsilon\delta_1\theta$ and one with coordinates $\theta+\varepsilon\delta_2 \theta$. In the limit of small $\varepsilon$, it seems reasonable to think of $$ \sum_{ij}g_{ij}(\theta)\delta_1\theta^i\delta_2\theta^j $$ as a scalar product between the "vectors" $\delta_1\theta$ and $\delta_2\theta$.
Is this a sensible way to think about scalar products and tangent vectors on a Riemannian manifold? If not, what's wrong with it? Or if so, is the expression I've written down correct, and how can it be related to the usual definition of a tangent vector in terms of differential operators?