Question about cancellation of electric field Why can't I cancel the two opposite electric field shown below?
I already searched about this and saw the article that says "It can't cancel each other because they aren't in the same region." I don't get it.

 A: The comment by @QuantumMechanic addresses this to some extent, but I think the key point that you're missing which is responsible for your confusion is that an Electric Field is not like the "vectors" you are familiar with in introductory classical mechanics. Instead, the Electric Field is a vector field. This means that it is represented by a vector at every point in space. In other words, you should always talk about an Electric Field vector at a point (i.e. it is a local quantity).
Now, vectors do cancel out when they oppose each other at a particular point in space, but that isn't the case in the problem that you have considered. The vectors that point "upwards" are "above" the plate, and those that point "downwards" are "below" the plate.
So, using the idea of a vector field, you can "draw" the same system as in your question as shown below (blue arrows for vectors pointing "up" and the red ones for vectors pointing "down"). Of course, every point in space has such an arrow, and I can't plot every one of them, so I've only plotted a small number of arrows. It should be obvious in this representation why all the vectors "above" are independent of those below:
                              
Let's now come to the question you had in the comments: why do we "cancel" the contribution outside a pair of oppositely charged sheets? Well, you can make the same sort of "diagram" as above. Consider two oppositely charged sheets at $z=0$ and $z=1$. Blue arrows still represent fields pointing "up" and red ones represent fields pointing "down". The diagram is a little messy, but you should be able to see that in between the two sheets, at every point, you would have a vector pointing up no matter what. Therefore, at every point the vectors add up constructively.
                              
However, outside the two sheets, at every point in space there is one "up" and one "down" vector, meaning that at every point the vectors add up destructively, leading to no Electric Field anywhere.
Note: Of course, you might ask, what of the vectors exactly on the plate. Well, I feel at the introductory level it would be a good idea to not focus too much about what happens to an Electric Field exactly on a charge distribution, as the result will blow up. (You could ask a similar question of a point charge, for example: when $r=0$, in which direction does the Electric Field of a point charge "point"?) As @Andrew points out in the comments below,

[O]ften these singularities hide a lot of interesting physics (in this case, the fact that any real plate has a thickness).

But such things are best left for a little later. In more advanced courses on Electromagnetism you will see that the Electric Field is indeed discontinuous over such boundaries, so it goes from pointing "up" to "down".
