STEP 01 · INPUT
Text / symbols
Words are meaningful to people, but not directly computable as coordinates.
CORE CONCEPT · 01
When text becomes geometry.
A language model cannot calculate directly with words. Vector representations turn symbols into coordinates—so relationships, directions, distance, and similarity can become measurable.
Vector field / 2D projection● DRAG TO EXPLORE
Direction + magnitude · v = (3, 2)Drag the glowing endpoint ↗
01 / The big picture
One journey, from symbols to relationships
Vectors are the bridge between text and mathematics. First we need numeric representations; then we learn how their dimensions and geometry let us compare them. Embeddings build on this foundation by learning useful representations of meaning.
STEP 02 · ENCODE
Numeric vector
A sequence of numbers gives each representation a place in a mathematical space.
STEP 03 · GEOMETRY
Direction & distance
Dimensions define axes; vector operations describe length, direction, and separation.
STEP 04 · COMPARE
Similarity
Measures such as cosine similarity compare vector directions. Embeddings use this idea to represent learned relationships.
02 / Live example
Move a vector. Watch the mathematics respond.
This small 2D example is a teaching model—not a real language model embedding. Adjust the coordinates or drag the endpoint in the hero diagram. Both views stay synchronized as length, angle, and cosine similarity update. In real systems, vectors may have hundreds or thousands of dimensions.
Vector controls
Vector A is the cyan arrow. Vector B is the violet arrow.
Try rotating B toward A. The cosine score approaches 1 when the directions align, 0 when perpendicular, and −1 when opposite.
Length A3.61
Length B3.61
Cosine0.98
cos(A,B) = (A · B) / (|A| |B|)
03 / Why this matters for AI
The embedding bridge
An embedding is a learned vector representation. Instead of assigning arbitrary coordinates by hand, a model learns numbers that help capture useful patterns from data. Geometry gives us tools to inspect and compare those representations.
Human-readable input
A sentence or token starts as symbols.
“king”“queen”“car”
The words themselves are not coordinates.
Learned numeric representation
An embedding model maps each item to a vector of numbers.
[0.21, −0.08, …][0.18, −0.02, …]
The short arrays are illustrative. Real embeddings are higher-dimensional; similarity is useful but not a perfect definition of meaning.
04 / Go deeper
Explore the subtopics
Build from the basics of numerical representation into vector geometry and the role vectors play in language models.
01Why Text Becomes Numbers
Why numerical representations are needed before mathematical operations can happen.
ESSENTIAL 02One-Hot EncodingA simple sparse representation: one active position among many zeros.
ESSENTIAL 03Sparse vs Dense VectorsCompare mostly-zero representations with compact, information-rich vectors.
SUPPORTING 04Vector DimensionsWhat it means for a vector to contain two, three, or many coordinates.
ESSENTIAL 05Vector SpaceThe mathematical space in which vectors can be represented and compared.
SUPPORTING 06DistanceMeasure how far apart two vectors are under a chosen metric.
SUPPORTING 07SimilarityCompare representations to estimate how closely they align or relate.
ESSENTIAL 08Dot ProductCombine corresponding coordinates to measure alignment and scale.
SUPPORTING 09Cosine SimilarityCompare vector directions while accounting for their magnitudes.
ESSENTIAL 10Static vs Contextual RepresentationsWhy a token's representation can vary with its surrounding context.
SUPPORTINGNext: learn how vectors can carry meaning
Embeddings build on vector representations. Open that topic after you're comfortable with coordinates, dimensions, and similarity.