Quick chemistry lesson:

Here's the structure of acetic acid: http://upload.wikimedia.org/wikipedia/commons/f/fd/Acetic-acid-2D-flat.png

Here's the structure of acetoacetic acid: http://upload.wikimedia.org/wikipedia/commons/d/d4/Acetoacetic_acid.png

First, note that the rightmost carbon of acetic acid, has a double bonded O and an OH attached to it. In chemistry terminology, this is usually written as COOH and a COOH group is an acid (carboxylic acid).

Look at the acetoacetic acid structure, you'll see a COOH on the left most carbon (it's not written out with a C here because they used shorthand where all unlabelled intersections of lines are assumed to be a carbon. So that COOH group makes it an acid. Then on the other end, there's a C double bonded to an O and a CH3 group. That's your ketone. Basically a C double bonded to an O is a ketone group (but it's not considered a ketone when there's the other OH group, you could say that the acid takes precident).

The reason you use ketostrips is to know if there are ketones in your urine. They specifically chose to measure acetoacetic acid, probably because it's an easy measurement, and use that as a proxy for all types of ketones present. But since acetic acid and acetoacetic acid have completely different structures, there will be no mixup in the measurement.

Also, I'm sure that the acetic acid you eat (as vinegar) gets broken down pretty fast in your digestive system, and doesn't last long as acetic acid in your body.

To answer your main question, I doubt it has any effect on ketosis. Your body will burn sugar when it's available. If it's not available, your body will turn fats into ketones and burn them. Having vinegar doesn't change the amount of sugar in your body, so if you're still low on sugar, you'll be burning ketones.

Think of it this way, your body will burn sugar or ketones with sugar taking precedent. This doesn't mean that sugar is the better fuel, ketones actually are the better fuel, but that sugar is so toxic you body will burn that fist just to get it out of your system.

I've never heard of vinegar used to help assist with ketosis. Acetic acid's not the same as acetoacetic acid. It could be there's some sort of conversion in the body, but it would mean that you're converting vinegar to ketone bodies, and not fat to ketone bodies. ie. false positive.

Side note, in re. coconut oil. It breaks down easily in the body, but also runs into a similar issue IMHO. ie. you get ketone bodies, but you're not necessarily breaking down bodyfat for it. So possible false positive on the strips. However, unlike vinegar, at least you're getting the liver used to processing more fat into ketone bodies.

In Adkin's earlier books he used to advise against vinegar in any major amounts because it raised the likelihood of yeast infections for men and women.

Unfortunately I don't have the book anymore and can't really find an online reference, but that's what I remember.

That being said, I am a low-carber Paleo and have been for quite some time, and I do enjoy cooking with vinegar in broths, stews, and roasts.

Vinegar is supposed to have some impact on abdominal fat, but I don't think/know if there's any impact on ketosis.

Just so you are aware, there is no need to continue testing your urine once you have gone into Ketosis, in fact, the farther into the future you get, the less ketones will show up in your urine. If your not downing carbs, your in ketosis, period. Even if you carb 'reload' you can get back into ketosis pretty quick, while having your muscles full of glycogen, just as soon as your liver dumps its gyclogen and your body clears it, bang, your back in ketosis (as muscle glycogen is locked and loaded for future use).