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A linearly polarized plane wave, say polarized along the x-axis, in a homogeneous medium has its electric field given by $$\mathbf E(x,y,z) = (A e^{ik(ay+bz)}, \ 0 \ , \ 0)$$ where $A,a,b$ are real constants, with $a^2+b^2=1$, and $k = 2\pi/\lambda$.

Now if a vector field has only one non vanishing component, this component equals its norm, which by definition is a scalar, right? So is such a plane wave field a scalar field?

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A scalar would be invariant under rotations. A non-zero vector is not, regardless of how many non-zero components it might have in any particular basis.

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