An object sits on an inclined plane. The weight of the object will have a normal and parallel component. I always thought that the reaction of the plane was simply the negative of the normal component of the weight.
Similarly, an object swings on a pendulum. The weight of the object can be decomposed into a radial and tangential component. I assumed that the reaction (tension) of the string must be the negative of the radial component.
But several examples have made me doubt my assumptions. One exercice in my textbook involves determining the reaction on a mass as it slides down a parabolic surface, as a function of $\theta$, the angle the vertical makes with the surface at any given point. Under my assumptions about reaction force, the answer is so trivial as to make the exerice pointless (since they give you $\theta$), but instead the exerice launches into a lot of complicated reasoning involving the Frenet-Serret base and comes out with a very different answer to just $-mg\ sin(\theta)$.
Also, consider the second example I gave. If the object is swinging in a circle (to make it even simpler, say it has uniform circular motion), then it must have radial acceleration (centripetal, specifically). But that's impossible if the reaction is exactly the negative of the radial weight.
But then... how can I determine a reaction force? In the absence of the simple rule I gave in the first paragraph, what else is there? Can it be done without making assumptions about the motion (eg. if you suppose the motion of an object is uniform circular, you have a formula all ready for the centripetal force and may thus be able to deduce the reaction force).