Choosing a vector difference function

I would advise those reading too much into this article not do so. Using vectors for theory to do basic statistics is likely to confuse matters for you. Vectors, though important in statistics, are primarily used to describe spaces, be they 3d or otherwise. If you understand how this translates to statistics, then you probably don't need the ideas in the article. If you don't, then this article is plainly misleading.

There are far better methods than the one linked in the article.

You can train a covariance matrix such that you can get a better distance metric. Particularly, you would use the Mahalanobis Distance:

http://en.wikipedia.org/wiki/Mahalanobis_distance

For classification tasks, there are two good ways of training a covariance matrix for distance metrics: neighborhood components analysis and large margin nearest neighbors.

The effect in the article is just a particular quirk of using the euclidean distance. You could, for example, get the same result by using a 1-norm distance.

This is usually called a "Norm" (or "Vector Norm").

http://en.wikipedia.org/wiki/Norm_%28mathematics%29