The state of matter with particles vibrating in a fixed position is solid.

Here's why:

* Solids have a definite shape and volume. This is because the particles are tightly packed together and held in place by strong forces.

* Liquids have a definite volume but not a definite shape. Their particles are closer together than in gases but can move around more freely.

* Gases have neither a definite shape nor volume. Their particles are far apart and move randomly.

So, while all states of matter have particles in constant motion, only in solids are the particles restricted to vibrations around fixed positions.