Why is the Length Element in Ampere's Law Perpendicular to the Current?

[Nway]

It does not have to be in any way. It is up to you to choose the integration path according to the problem you try to solve. $d\vec{s}$ is the path element for any path, no matter what is the problem to be solved. You pick the integration path to solve the problem, usualy to solve it in the easiest way. You need to understand how the laws used (here Biot-Savart and Ampere) work in order to understand how do you choose the integration path. Once you do that, you don't have to make any choice about $d\vec{s}$. It is tangent to the chosen path.

Can $\vec{ds}$ also be though of as the displacement of a positive test using the right hand rule?

 

[nasu]

Can $\vec{ds}$ also be though of as the displacement of a positive test using the right hand rule?

I don't know what you mean by this.

 

[SammyS]

Can $\vec{ds}$ also be though of as the displacement of a positive test using the right hand rule?

No, $\displaystyle \vec{ds}$ cannot be thought of as the displacement of a positive test charge, using the right hand rule?

In using Ampere's law, you are integrating the magnetic field along a closed path.