I recently had to solve the following problem:
A force $F=10\hat x- 8\hat y$ is applied to an object that is constrained to travel towards increasing values of x along the path defined by $y=x^2$ and $z=0$. Find the component of $F$ that is tangent to this path at the point $(2,4,0)$.
Attempt: I used Pythagorean's theorem to $ F=12.81 N$ since the $x$ and $y$ components are given. I just wasn't sure if this is what the questions is asking for, or is there something else I have do do?
Edit. More generally, suppose that one is given a curve and a force field on the plane. How would one go about computing the component of such a force field in the direction tangent to the curve at a given point?