The following is a (heavy) paraphrase from page 19 of the book Ordinary Differential Equation by Arnol'd
Consider the equation $x'(t) = v(x(t)), x \in \mathbb{R}$. The RHS defines a phase velocity vector field: a vector $v(x)$ is attached at the point $x$. Assume that $v(x)$ is continuous and never vanishes. The tangent of the angle between our field and the x-axis equals $1/v(x)$.
My question is is that why is the tangent of the angle between a general vector field and the ordinate axis equal to $1/v(x)$? The author provides no derivation for his claim, and I have been unable to find a derivation for the angle from the Internet.