I am watching the video series "Multivariable Calculus" by Prof. Herbert Gross. In [Part III, Lecture 2](https://youtu.be/UGKL1wHouho?t=1759) it is mentioned that if you have some function: $w = f(y)$, the tangent line to that curve at a point $P\_0$ can be written in vector form using Cartesian Coordinates and more specifically one vector that represents the tangent line would be the following:
$$\vec{v\_1} = \vec{j} + \frac{dw}{dy} \vec{k}$$

where $\vec{j}$ is a unit vector in the y-direction and $\vec{k}$ is a unit vector in the w-direction (the derivative is evaluated at $P\_0$).

I can't seem to grasp why this would be the case. How is this vector obtained step by step?