Tangent Vectors to Fibers of the Hopf Map

Question

So I'm viewing the Hopf map as a map from $\mathbb{C}^{2} \to \mathbb{R} \oplus \mathbb{C}$ by $(z,w) \to \left(\frac{1}{2}(|z|^{2}-|w|^{2}),z\bar{w}\right)$. I've concluded that the fiber including $(z,w) \in \mathbb{C}^{2}$ is the set of points $(e^{i\theta}z,e^{i\theta}w)$ for $0 \le \theta < 2\pi$. What I'm having trouble with is the following: I was told (in a text) that $i(z,w)$ is tangent to the fiber including $(z,w)$. As I understand it, this means $i(z,w)$ is a vector in the tangent space of each point in the fiber of $(z,w)$. Is this understanding correct, and if so how do I conclude that $i(z,w)$ is actually tangent to the fiber? Is there a fast way to check this with inner products (I'm viewing this problem from the category of Riemannian manifolds)?