Position vector in curvilinear coordinates

[Jhenrique]

The position vector $\vec{r}$ in cartesian coordinates is: $\vec{r} = x \hat{x} + y \hat{y}$, in polar coordinates is: $\vec{r} = r \hat{r}$. But, given a curve s in somewhere of plane, with tangent unit vector $\hat{t}$ and normal unit vector $\hat{n}$ along of s, exist a definition for the position vector in terms of $\hat{t}$ and vector $\hat{n}$?

 

[Simon Bridge]

Short answer: kinda.
Coordinate systems are maps - you can make any kind of map you like.
Some coordinate systems are more useful than others. Note: in order to specify an arbtrary position in 3D space, you usually need three numbers.

So you could say - go distance s along this particular curve, rotate by so much angle a, then follow the normal away from the curve a distance d. So your position vector would be r=(s,a,d) ... it may be that a particular point P will have more than one representation, also the order that the components are followed will probably matter.

 

[Chestermiller]

The position vector $\vec{r}$ in cartesian coordinates is: $\vec{r} = x \hat{x} + y \hat{y}$, in polar coordinates is: $\vec{r} = r \hat{r}$. But, given a curve s in somewhere of plane, with tangent unit vector $\hat{t}$ and normal unit vector $\hat{n}$ along of s, exist a definition for the position vector in terms of $\hat{t}$ and vector $\hat{n}$?

Sure. Just resolve $\vec{r}$ into components in the $\vec{t}$ and $\vec{n}$ directions:

$$\vec{r}=(\vec{r}\centerdot \vec{t})\vec{t}+(\vec{r}\centerdot \vec{n})\vec{n}$$

Chet