Why do the planets in the Solar system stay in the same orbital plane? An earlier question addressed why all planets formed in the same orbital plane, but how is this angle maintained? What prevents the planets from taking on a different orbital plane?
Answer to the NEW question:

the Angular Momentum Conservation Law states that, for any moving body, its angular momentum does not change unless you exercise an external force different from the central force.

For an orbiting body like a planet, this means that Sun's gravity, being the central force, does not modify Angular Momentum, but any other external force will do.

Examples of external forces are collisions or the forces made by Jupiter on another planet, or by Neptune on Pluto.

After the Solar System was formed, these external forces are quite small, and thus does not change greatly the Angular Momentum of any major body. But you can see how passing near a body can alter a comet's orbit.

Moreover, the external forces made by bodies that are in the same plane as an orbiting body does modify the value of its Angular Momentum, but not the direction. This causes that  the orbiting body changes its orbit but can not not change planes.

So if you add small forces from objects in the same plane, you end up with no changes to planes.