Given a direction vector, I would like to know if a step taken along it would be past the root of a vector-valued function.
In the 2D case, this is relatively simple: if $f(x_1) * f(x_2) < 0$ then we can conclude that we have shot past the root as you can see on the following image.
Now, how do I extend this idea to the $n$-dimensional case? The option to use the Jacobian to obtain this information is also open. All inputs are welcome, thanks in advance!
After considering the 2D case e.g. $f : \mathbb{R}^2 \mapsto \mathbb{R}^2$, I was able to conclude that the multidimensional test for crossing of a root could well be simply $f_i(\vec{x}) * f_i(\vec{x} + \vec{d}) < 0$ for all $i = 1...2$. This can be visualized as follows (the crossing of a root is indicated by the arrows for each quadrant)