Under what specific conditions would 'dot product' be a more appropriate vector similarity metric than 'cosine similarity' for retrieval in a RAG system?

Dot product is a mathematical operation that takes two vectors and returns a single scalar number representing their similarity, incorporating both their directional alignment and their magnitudes, which are their lengths. Cosine similarity, conversely, measures only the directional alignment between two vectors by first normalizing their magnitudes to unit length before calculating their dot product. Therefore, cosine similarity focuses purely on the angle between vectors, disregarding how long or short they are. Dot product becomes a more appropriate vector similarity metric than cosine similarity for retrieval in a Retrieval Augmented Generation (RAG) system under specific conditions where the magnitude of the embedding vectors is intentionally designed or trained to encode additional meaningful information beyond mere semantic direction. This means the embedding model explicitly assigns semantic relevance to vector length. One primary condition is when vector magnitude represents the importance, salience, or frequency of the concept, term, or entity that the vector embeds. For instance, an embedding model might produce longer vectors for concepts that are central, highly important, or very frequently discussed within the corpus, while generating shorter vectors for peripheral, less important, or rare concepts. In such a scenario, using dot product allows the retrieval system to factor in this magnitude-encoded importance. A query vector with high magnitude matching a document vector with high magnitude through a strong directional alignment would result in a higher dot product score, indicating a superior match not just in topic, but also in perceived significance or centrality. This contrasts with cosine similarity, which would treat a match with an important, long vector identically to a match with a less important, short vector, provided their directions are the same. Another condition is when vector magnitude signifies the specificity or generality of the embedded information. A highly specific or detailed concept might be represented by a longer vector, whereas a more general or abstract concept might have a shorter vector. Dot product would then favor matches between specific query vectors and specific document vectors, or general query vectors and general document vectors, capturing this nuance. Furthermore, if the embedding model is designed such that vector magnitude reflects confidence or certainty in the embedded representation, dot product would incorporate this. A longer vector might imply higher confidence in the embedding's accuracy or the certainty of the information it represents. Finally, dot product is appropriate when the embedding model's output vectors are not normalized to unit length, and their varying magnitudes are specifically intended to convey semantic meaning as described above. If the vectors are already normalized to unit length (a common practice with many modern embedding models), then dot product and cosine similarity produce the same ranking of results for retrieval, as the magnitude terms in the dot product calculation become one. However, the question specifically asks for conditions where dot product is *more appropriate*, which implies situations where the magnitude itself carries distinct, valuable information for retrieval that cosine similarity would discard.