Non-commutative distance formula
Take for example the Euclidean distance L2, defined by:
$$L_2(x,y) = \left(\sum_{i=1}^{d} (x_i-y_i)^2\right)^{1/2}$$
where $d$ is the vector dimension. You can easily add a term $\alpha \in (0,1)$ and put more weight on the first term, for example:
$$L_2(x,y) = \left(\sum_{i=1}^{d} (\alpha x_i-(1-\alpha)y_i)^2\right)^{1/2}$$
That will certainly be non-cummutative and basically it can be interpreted as putting more weight on the 'reference' vector.
Now, if you ask me, what would be a good use of it, I hardly understand why you would use a non-commutative distance or non-symmetric distance. First of all, it will not be a distance in mathematical sense, since symmetry is required by definition, secondly, many algorithms works with symmetric distances and can't be adapted easily.