Importance of Newton's Third Law of Motion

The third law states that action and reaction are always an equal-and-opposite PAIR of forces, acting along the same line, and -- crucially -- always acting on two DIFFERENT bodies, never on the same body.

If a body x exerts a force on body y, conventionally written $\vec{F}_{yx}$ (read as 'force on y due to x'), then body y exerts an equal and opposite reaction force $\vec{F}_{xy}$ back on body x. Body x experiences ONLY $\vec{F}_{xy}$ and body y experiences ONLY $\vec{F}_{yx}$; even though $\vec{F}_{xy}=-\vec{F}_{yx}$ in magnitude and direction, they can never cancel each other in any calculation of the net force on a single body, because each one acts on a DIFFERENT object. This is the single most common point of confusion with the third law and worth remembering as a standing rule: action-reaction pairs never both appear in the free-body diagram of one object.

Action-reaction pairs need not involve physical contact at all. Two magnets repelling each other, or the Earth and the Moon (or the Earth and the Sun) attracting each other gravitationally, are equally valid action-reaction pairs even though the bodies never touch. …