I apologize if this question is too trivial, but I don't have anyone else to ask and I don't understand it.
In class, we solved an example. The question was "Find the parametric equation of the line tangent to the curve $\vec{r}(t)=t\overrightarrow{i}+e^{-t}\overrightarrow{j}+(2t-t^2)\overrightarrow{k}$ at the point $(0,1,0)$."
We differentiated $\vec{r}(t)$ and plugged in $0$ to get a direction vector. Using the direction vector and the base point, we found the parametric equation of the line.
However, how can a function with 3 dimensions have a tangent "line"? Shouldn't it be a plane? Couldn't any tangent line be rotated with respect to the base point to create a plane? I am imagining a shape like a windmill.
What we did to solve the question sounds reasonable, but I don't think I really understand it. It seems like there must be something that makes the line we found more "special" than all the other lines in the plane that I (tried to) describe.