On flat ground, "perpendicular to the surface" and "straight up" are the same direction — so it never mattered that normal force problems used the ordinary vertical/horizontal axes. On an incline, the surface itself is tilted, so normal force and friction both point along tilted directions.
Gravity still points straight down in the real world — it's your tilted axes that make it look "diagonal." Split it into a component parallel to the ramp and one perpendicular to it, using the incline angle θ.
The parallel component (mg sinθ) is the piece of gravity actually trying to slide the object down the ramp. The perpendicular component (mg cosθ) is the piece pressing the object into the ramp's surface.
Drag the incline angle and watch weight split into its two components live — along with the normal force and friction that respond to it.
Drag θ down to 0° — the parallel (cyan) arrow should shrink to nothing. Drag it up toward 80–90° — the normal (green) arrow should shrink toward nothing. That's the sanity check from the Quick Reference, live.
The object doesn't accelerate into or out of the ramp's surface — it stays on it. That means the perpendicular direction is in equilibrium, exactly like Lesson 2-1-2's balanced forces: the normal force must exactly cancel gravity's perpendicular component.
With the perpendicular direction settled (normal force cancels gravity's perpendicular piece), everything interesting happens along the parallel direction. If the object is sliding, friction opposes that sliding — which usually means friction points up the slope while gravity's parallel component points down it.
A 4 kg block sits on a 25° ramp with μk = 0.25. Find its acceleration as it slides down.
Swapping sine and cosine is the single most common incline error — writing mg cosθ for the parallel component and mg sinθ for the normal force instead of the other way around. Both formulas involve the same two functions, and it's easy to grab the wrong one under time pressure.
There's one especially elegant special case: the exact angle where an object is right on the verge of sliding. At that angle, gravity's parallel component exactly equals maximum static friction — and when you set mg sinθ = μs mg cosθ, the mg cancels entirely, leaving tanθ = μs. Project 2-2-4 is built directly on that relationship.