Sometime ago I had posted the What is the origin of "normal" in normal coordinates and normal modes? in math & sci history problems. Nobody was sure for the reason for using the word normal there - it was an interesting discussion though. The concept of normal coordinates is transiently brought in vibrational spectroscopy in undergraduate textbooks but rarely explained in detail. Certainly, the concept is borrowed from mechanics.
How can we explain the concept of normal coordinates? What is the need of a new set of coordinates if x, y, z can show also the displacements of an atom from its mean position? Let us take the example of a non-linear molecule like water.
Edits More specifically, how would we mathematically define a normal coordinate for the oxygen(s) and hydrogen during a normal mode vibration of water molecule? Or a "normal coordinate" is a global displacement coordinate for the whole molecule for a given normal mode vibration. Water has 3 IR active modes. Do we have only 3 normal coordinates $q's$ for water? Wikipedia shows a simplistic picture of a normal coordinate $q$ as a measure of displacement but from little bit of reading Herzberg's other specific monographs by Wilson, the concept of normal coordinate does not seem that trivial.