Is superposition principle equally valid for both scalar and vector quantities? I was wondering if superposition is even applicable to scalar quantities because scalars should simply add up to give final result as there is no sense of direction. Do we have any example of a vector quantity that does not follow superposition principle?
 A: Superposition principle is valid if and only if your equations are linear (it doesn't matter if they are scalar or vector quantities). For example if we have a scalar field $\phi$ that satisfy the equation
$$
\partial^\mu \partial_\mu \phi + \lambda \phi^3 = 0
$$ 
Then the superposition principle is not valid, since if you have two fields $\phi_1$ and $\phi_2$, the resulting effect of both together is not $\phi_1 + \phi_2$, since $\phi_1 + \phi_2$ is not solution of the equation above. It happens that a lot of equation in physics are linear, so superposition principle is applicable (like in Maxwell equations for example), but there are some (like Navier- Stokes equations or Einstein field equations) which are not, so its a lot more difficult find exact solutions.
A: Hector's answer seems good to me. But since you ask, I can give you an example of a vector field which does not superpose.
The Poynting vector in electromagnetism is given by ${\bf S} = {\bf E} \times {\bf H}$, but if two electromagnetic fields are superposed, their Poynting vectors do not simply sum to give the Poynting vector of the combined field.
You have to find the combined E- and H- field and then determine the resultant Poynting vector.
