Norm induced by the inner product

Question

Fix $S^1 \times I$ where $I \subset \mathbb{R}$ with coordinates $(\theta,t)$. Let $ds^2 = dt^2 + f(t)^2d\theta^2$ be the metric (first fundamental form). A basis of tangent vectors on $S^1 \times I$ is given by $\partial_\theta$ and $\partial_t$. These are orthonormal if $\|\partial_\theta\| = \langle \partial_\theta, \partial_\theta\rangle^{1/2} = 1$. Now I have a question: the metric is the one mentioned above, so the norm should be the one that is induced by the metric. Moreover, $\partial_\theta$ can be obtained from $\partial_\theta = (-\sin(\theta),\cos(\theta),0)$, right? Now If I wanted to compute the norm of this vector under $ds^2$ how would I do it? because I have three entries, as seen by the injection in $\mathbb{R}^3$ but the metric is only considering the two coordinates $\theta$ and $t$.