Use Stokes's Theorem to show that
$$\oint_C=y\,dx + z\,dy + x\,dz = \sqrt{3}\pi a^2,$$ where $C$ is the suitably oriented intersection of the surfaces $x^2+y^2+z^2=a^2$ and $x+y+z=0$.
We get that $$\textbf{F} =y \textbf{i} + z\textbf{j} + k\textbf{k}$$ and $$\textbf{curl F} = -(\textbf{i} + \textbf{j} + \textbf{k})$$ and a normal vector $$\textbf{n} = \pm\frac{1}{\sqrt{3}}(\textbf{i} + \textbf{j} + \textbf{k}).$$
Cool. But how do I know which direction the normal vector should be in, should it be positive or negative? Sure, if we put in the normal vector as negative, we get the right answer, but how do I know the orientation of the boundry $C$?
Why does the value change if I put in the positive normal vector? Is it possible to get the same result when using the positive normal vector?
I know the fact that if traversing the boundry, the normal vector should be to left of the "viewer". But let's say we get zero information on what orientation the boundry is in, how do we know what direction the normal vector should point in then?
If you do not fix orientation the line integral is not uniquely defined. The definition of the line integral is independent of parametrization but dependent on orientation.
For the Kelvin-Stokes theorem the curve should have positive orientation, meaning it should go counterclockwise when the surface normal points towards the viewer. Once you've picked that convention you can use the normal vector to control the orientation. Flipping the normal vector changes the orientation. It's an arbitrary choice similar to the cross product which is again dependent on orientation.
Since the problem didn't specify the orientation (says "suitably oriented") they probably wanted the person trying to solve it to pick whatever direction of the normal gets the right result.