---
title: "Embeddings, similarity, dimensions, and vector identity"
chapter: "08"
---

# Embeddings, similarity, dimensions, and vector identity

An embedding turns content into a numeric vector whose direction represents
learned semantic relationships. Similar meanings tend to be near each other
in that model's vector space.

## Similarity

Cosine similarity compares direction:

`cos(a,b) = (a · b) / (||a|| ||b||)`

Dot product is equivalent for unit-normalized vectors. Euclidean distance can
work with the index and normalization chosen. Never compare raw threshold
numbers across different models or distance functions.

## Vector identity

Store with every vector:

- embedding provider, exact model, revision, and dimensions;
- normalization behavior;
- source document/version and chunker version;
- content hash, tenant, language, and authorization metadata;
- creation time and ingestion run.

Index and query content must use the same embedding space. Changing dimensions
or model requires a parallel index, re-embedding, evaluation, traffic switch,
and rollback window.

## Dimension trade-off

More dimensions may improve nuance but increase storage, memory, network, and
index cost. Benchmark the actual corpus at 768/1536/3072 or model-appropriate
sizes. “Largest” is not a universal quality decision.

## Multimodal spaces

Gemini Embedding 2 can map text, images, audio, video, and PDFs into one space.
Evaluate cross-modal retrieval separately: text-to-image recall is a different
task from text-to-text question answering.

## Privacy

Embeddings can leak membership or meaning and remain personal data when tied to
people. Apply retention, encryption, access control, deletion, and tenant
isolation to vectors and source content.

## Feynman check

An embedding is a map coordinate for meaning. Coordinates from two different
mapmakers cannot be mixed and expected to point to the same street.