Embeddings are everywhere in modern AI, but the word "embedding" is doing a lot of different jobs. A token embedding is a learned row in a vocabulary matrix. A contextual embedding is a hidden-state vector whose value depends on the surrounding sequence. A sentence or document embedding usually requires another step => some pooling or model-specific readout that turns variable-length representations into one fixed-size vector. These are related ideas, but they are not interchangeable. The geometry is learned too. There is nothing inherently semantic about a vector just because it has 768 or 1,024 dimensions. The training objective has to shape the space so that useful pairs score well and unrelated examples separate. That is why contrastive training, hard negatives, pooling strategy, normalization, and the similarity function matter so much in modern retrieval systems. This also explains several production bugs that look harmless at first. Cosine similarity and dot product are not generally the same, though they coincide for unit-normalized vectors. Changing the embedding model while keeping the same dimensionality does not preserve the coordinate system. Ignoring query/document prefixes can change results for models trained asymmetrically. Changing preprocessing or pooling can invalidate an existing index even when every vector still has the expected shape. I put together a technical handbook that works through embeddings from first principles => token lookup tables, contextual representations, pooling, cosine/dot/L2 geometry, contrastive learning, dense retrieval, hard negatives, Matryoshka representations, multilingual and multimodal embeddings, and the implementation details that matter when these vectors are used for search and RAG. The mental model I kept coming back to while writing it is simple => an embedding is useful geometry learned for a purpose. To understand one, you need to know what was mapped, how the vector was produced, what objective shaped the space, and how those vectors are compared. Sharing the handbook here: