How would one approach a problem of finding intersection points between a line $\vec l = \vec S + d \vec t$ and vector of the form $$\vec v = \begin{pmatrix} x \\ y \\f(x, y) \end{pmatrix}$$
I am mainly looking for fast numerical methods. If anyone would take the time to explain through example of $f(x, y) = sin(x) + sin(y) $
If there is a more general form to solve intersections between the line and vectors of form $\vec v'= \begin{pmatrix} f_1(t) \\ f_2(t) \\f_3(t) \end{pmatrix}$ then I would love to know that as well. To be exact I am looking for the first intersection (intersection with the lowest distance d in direction $\vec t$) Hope the question is not too vague. If it is please tell me about the constraints needed. Any reference or keyword helps. Thank you.
EDIT:
I am thinking that we can write $\vec l$ as a combination of its parts and then look at it as set of equations. Meaning: $$\vec l = \begin{pmatrix} \vec S_x + d \vec t_x \\ \vec S_y + d \vec t_y \\ \vec S_z + d \vec t_z\end{pmatrix} \rightarrow \vec l = \vec v \rightarrow \begin{matrix} \vec S_x + d \vec t_x = x \\ \vec S_y + d \vec t_y = y \\ \vec S_z + d \vec t_z = f(x, y) \end{matrix}$$