Chapter 2: Representing Meaning Using Vectors

2.1 Introduction

Computers process numerical information. Therefore, language must be converted into mathematical form before machines can analyze it.

2.2 What is a Vector?

A vector is an ordered list of numbers representing features.

Example:

King → [0.2, 0.8, 0.5]

Queen → [0.21, 0.79, 0.52]

2.3 Geometric Interpretation

Vectors can be visualized as points in multi-dimensional space. Distance between points measures similarity.

Similarity Measures

• Euclidean Distance
• Cosine Similarity

2.4 Word Relationships

Vector arithmetic captures semantic relationships:

King − Man + Woman ≈ Queen

2.5 Embeddings in Modern AI

Large language models convert entire sentences into high-dimensional vectors known as embeddings.

2.6 Applications

• Search engines ranking results
• Chatbots predicting responses
• Recommendation systems
• Spam detection

2.7 Extended Example

Suppose we map the words Dog, Cat, Lion, and Car into 2D space. Dog, Cat, and Lion cluster together (animals), while Car is positioned far away.

2.8 Chapter Summary

Vectors allow computers to mathematically represent meaning. Similarity in vector space corresponds to similarity in language.