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The completely general way to represent a polarized wave is via the unit helicity vectors :)
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unit helicity vector?
both look linearly polarized along line y = -x, if you add the vectors.
same magnitudes, same phase.
and though the (kx - wt) formulation in example 2 looks somewhat odd to me, shouldn't matter
Try not to get lost in all the stuff inside the cosine and sine functions.
In trigonometry we learn that cosine and sine are just bouncing back and forth between 1 and -1. So when we plug in the max and min values of each function we can start to paint a picture of whats happening here. [NOTE: gotta remember that i and j stand for x and y respectively, they are components of the total E vector.]
When you plug 1 in for both cosines in the first equation you see the x (or "i") component remains positive while the y (or "j") component becomes negative. When you plug -1 in for both cosines (again just the first eq) x becomes a negative component and y a positive component. This helps show why the electrical wave is linearly polarized along the line y=-x.
To more rigorously prove this I would suggest looking up JONES MATRICES and using that mathematical process in the future.