I have got a problem to devise a distance metric to get the similarity measurement of vectors. Someone suggested me to use dot product, which seems to me the same as the Cosine similarity metric; however in Wikipedia (Cosine Similarity), it mentioned Cosine similarity is not a proper distance metric as it does not have the triangle inequality property and it violates the coincidence axiom (the proper distance metric should satisfy the four conditions (distance metric)).
My questions are:
What are the proper distance metric? Please name some examples.
Are Dice's coefficient and Jaccard index proper distance metric?
Are there any disadvantages of using dot product? (One of the reasons for the popularity of dot product is that it is very efficient to evaluate).
Thanks a lot. A.