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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?

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    $\begingroup$ I might be way off base here but, as far as I know, vectors live in a vector space and, by definition, a vector space is linear. See a related question here: math.stackexchange.com/q/479095 $\endgroup$ – Alfred Centauri Oct 24 '14 at 2:46
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    $\begingroup$ @AlfredCentauri: The linearity of vector spaces as a matter of definition simply means that we have a clear definition of what it would mean for superposition to hold for a vector field. It doesn't mean that it does hold. For example, sound waves can be describes as a vector field, if you use a vector to describe the displacement. But sound waves only approximately superpose. $\endgroup$ – Ben Crowell Oct 24 '14 at 5:34
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    $\begingroup$ @BenCrowell, correct but, as in, e.g., Fourier analysis, we can write the solution as a linear superposition of weighted basis functions. I concur that non-linear vector valued equations do not allow superposition of solutions to make another solution but it isn't (wasn't?) clear to me if this is Swami's question or not. $\endgroup$ – Alfred Centauri Oct 24 '14 at 11:10
  • $\begingroup$ If you mean "vector" in the sense that mathematicians mean it -- or in the quantum-mechanics/wave-mechanics sense -- stuff that lives in a linear space -- then the superposition principle exclusively applies to vectors, and it doesn't make sense to ask if it applies to scalars, because scalars are just the things that scale vectors inside superpositions. $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 16 '18 at 5:06
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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.

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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.

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  • $\begingroup$ Great example Rob. $\endgroup$ – Swami Oct 24 '14 at 7:34

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