# Similarity metric not based on statistical correlation

I'm building a simple recommender algorithm and as a prerequisite to calculating a prediction, I need to assign a similarity metric between each two pair of users. The next step being calculating a weighted average between all users.

My problem with Pearson's or cosine is this:
A, B are users whose vectors represent some ranking they have made on 6 items in a scale of 5.
Let A = [1,1,1,0,1,1]
Let B = [5,5,5,0,5,5]
Then the similarity s is:

s(pearson)=1
s(cosine)=1


So basically there's a full correlation between A,B, although user A ranks low and user B ranks high, and the users aren't literally “similar”.

I would like to use a metric that relates how 'close' A,B are to each other.
I used the following algorithm:
1. for each item i both users A,B have assigned a value r to, I used
$${{q}_{i}}=\frac{\left| {{r}_{A,i}}-{{r}_{B,i}} \right|}{scale}$$ 2.
$$c(A,B)=1-\frac{\sum\limits_{i}{{{q}_{i}}}}{\sum\limits_{i}{{}}}$$
3. Then from a scale of 0 to 1 (1 being 'alike') c(A,B) is the similarity metric (c for closeness).

EDIT:
I just noticed that what I did is just a variation of a normalized Minkowski Distance with a dimension of r=1

I know it's a bit general, but is the metric c makes any sense in respect to the function I want it to serve?

More to the point, will I see any improvement in the prediction if instead of doing a weighted average on let's say 10 users with the highest s to the user I try to accommodate, I'll use the ones with the highest c?