Vinegar And Laundry

The internet claim:  "Just adding 1/2 cup vinegar to the final rinse will result in brighter, clearer colors and fluffier laundry"

The science: CH3COOH + H20 --> CH3COO- + H3O+
vinegar (acetic acid) + water --> acetate ion (which is a very weak acid) + Hydronium ion
The acidic solution is supposed to dissolve soap reside (alkalies) left by soaps and detergents allowing the laundry fibers to because more absorbent and fluffy.

An ion is a charged atom or molecule. It is charged because the number of electrons do not equal the number of protons in the atom or molecule. An atom can acquire a positive charge or a negative charge depending on whether the number of electrons in an atom is greater or less then the number of protons in the atom.
Warning: Never combine vinegar with chlorine bleach as this will result in harmful fumes. Do not use vinegar on silk, acetate, or rayon clothing. 

My claim: Adding vinegar to laundry is redundant and will not cut laundry costs if you are using home made detergent.  In the following reaction, I assume the baking soda is coming from your home made detergent.

The science: NaHCO₃ + HC₂H₃O₂ → NaC₂H₃O₂ + H₂CO₃
(baking soda) + (vinegar) → sodium acetate + carbonic acid
Carbonic acid is unstable and undergoes a decomposition reaction to produce the carbon dioxide gas:
H₂CO₃ → H₂O + CO₂
resulting in NaHCO₃ + HC₂H₃O₂ → NaC₂H₃O₂ + H₂O + CO₂
The carbon dioxide escapes the solution as bubbles like the foamy lava experiment you did in 3rd grade. A dilute sodium acetate solution remains after the reaction. The sodium acetate (which is a salt) may offer a tiny bit of grease-removing ability, but it’s not much. Probably not enough to really notice.  So, really, all you’re doing, when you combine baking soda and vinegar to make a cleaning agent, is cleaning with very neutralized water with just a tiny amount of salt in it.
That’s not going to get your clothes very clean, and it’s certainly not going to soften your fabrics much.