Question

how the electrons behave as both particle and wave . how it shows this property when it does not vibrate

Answer 1

The particles do not vibrate, but their location in space can be described mathematically as a wave of PROBABILITY of finding the electron at a given point. Quantum mechanics is difficult to summarize, I suggest reading a book like "Schroedinger's Cat" for a more detailed explaination.

Answer 2

Every particle or a wave has dual character. Refer De Broglie's equations. Every mass can be converted in wave and vice-versa.