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A plane has several quite different meanings. A plane isA tree, otherwise known as a SycamoreA flat surface, more often conceptual than physicalA tool for smoothing a piece of woodOne of the lifting surfaces of an airplane - usually the tail-plane (aka horizontal stabiliser) or the wing (sometimes called main-plane)A level of thought or existenceA short form of airplane (q.v.). Strictly this one should be written 'plane not plane.
An elevator on a airplane are tabs on the tail that control its up and down motion in the air
Yes if you put them "head to tail" and the head of the fourth one points to the tail of the first one the resultant is zero.
You can add vectors graphically, by drawing them head-to-tail. Algebraically, you can separate them into components (for example, in two dimensions, the horizontal and the vertical component), then add those.
It changes the profile of the wing and so changes the lift. If the same change is applied to both wings the plane will gain or lose height. If opposite changes are applied, one wing will drop, the other will rise and the plane will turn.