Change in correlation after change in distance measure between two vectors I have two vectors A and B for which I calculate one of 2 measures (say X): 1. Euclidean distance 2. Cosine similarity
I now correlate this measure X with another independent measure Y which too depends on A and B. Now, In the case when X is Euclidean distance, X and Y show a negative correlation whereas in the case when X is Cosine similarity, X and Y show a positive correlation (Here A, B and Y remain the same). Can you please help me in understanding how this can be given that the results are conflicting?
 A: When $A = B$, the euclidean distance will be zero, while the cosine similarity will be one. As $A$ becomes more dissimilar to $B$ (with respect to their orientation) euclidean distance grows while the cosine similarity decreases.
Cosine similarity measures the similarity of orientation. For example, if the orientation is same i.e. angle difference is $0$ degree then the cosine similarity equals $1$ and if the angle difference is $90$ degree then the cosine similarity is $0$. On the other hand euclidean distance, in a way, accounts for the change in both magnitude and orientation.
Now, if you choose your vectors $A$ and $B$ such that $\left\|A\right\|_2=\left\|B\right\|_2$, then as you increase the angle the angle between the two vectors from $0$ to $180$, the euclidean distance will grom from $0$ to $2\left\|A\right\|_2=2\left\|B\right\|_2$, while the cosine similarity will decrease from $1$ to $-1$.
If the measure $Y$ decreases as the angle difference between $A$ and $B$ increases then $Y$ and cosine similarity will have positive correlation and $Y$ and euclidean distance will have negative correlation. There can be many possibilities  for $Y$ but in all of them $Y$ will show the described behaviour which will justify the signs of the two correlations. While correlating with the euclidean distance, the change in value of $Y$ based on the change in the magnitude of the two vectors, will also a play a role in defining the correlation between euclidean distance and $Y$ but as long as the orientation difference plays a dominant role in the computation of $Y$ than the magnitude difference, the signs of the correlation of $Y$ with euclidean distance (-) and cosine similarity (+) will be justified.
