An ordered list
[0.2, −0.7, 0.4] is a three-dimensional vector. Real embedding models often produce hundreds or thousands of coordinates.
A visual introduction
Meaning
has geometry.
An embedding turns something—like a word, image, or song—into a list of numbers. Once meaning has coordinates, we can measure it, compare it, and search through it.
Start with a word01 / The object
An embedding is a position.
A model converts an input into a vector: an ordered list of numbers. Each number is one coordinate, and together they locate the input in a high-dimensional space.
Choose an input
A drink, usually warm, bitter, and energizing.
A simplified 8D vector
v(coffee)[0.82, −0.14, 0.61, 0.33, −0.48, 0.09, 0.74, −0.21]
Every possible position
A vector space is the mathematical world those vectors inhabit. Directions and distances inside it can encode patterns in the data.
One coordinate axis
Dimensions are learned together. A single axis rarely has a clean label like “warmth”; meaning is distributed across many of them.
02 / The geometry
Nearby vectors behave alike.
During training, the model organizes inputs by patterns of use and content. Items that appear in similar contexts tend to land near one another—even when their raw forms are very different.
Query point
Nearest neighbors
This 2D map is a projection. The real comparison happens in the model’s full-dimensional space.
semantic space · projected to 2D
A common measure
Cosine similarity asks: do they point in the same direction?
It compares the angle between two vectors and mostly ignores their length. A score of 1 means the same direction, 0 means perpendicular, and −1 means opposite directions.
cosine similarity0.88
03 / The search engine
A vector database searches by meaning.
Instead of looking for exact words, a vector database stores embeddings and efficiently retrieves the vectors closest to a query. This is called nearest-neighbor search.
01Embed query
[.31, −.08, …]02Scan index
≈ nearest vectors03Return matches
top k = 3Corner library
Soft chairs, silent rooms, and shelves of old books.
Garden bench
A shaded seat away from the busiest walking paths.
Small café upstairs
A calm second floor with window seats and low music.
A vector database is not the model. The model creates coordinates; the database stores and searches them.
Each record keeps its vector alongside an ID, metadata, and often the original text or image.
Specialized indexes avoid comparing the query with every vector, trading a little precision for enormous speed.
Applications use the nearest results for semantic search, recommendations, clustering, and retrieval-augmented generation.
The core idea
Encode.
Compare.
Retrieve.
Embeddings make meaning computable—not because any one coordinate explains it, but because relationships emerge from the geometry of the whole space.
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