Interactive Lab 02 — Vectors: Numbers with Meaning

1. A vector is a coordinate story

The vector [3, 4] is not automatically a movement, a point, a purchase, or a document. The numbers become meaningful when we decide what the coordinates represent.

point: (3,4) movement: 3 east, 4 north shopping: 3 coffees, 4 sandwiches text: 3 counts of word A, 4 counts of word B

2. Vector as point and arrow

x-coordinate: 3

y-coordinate: 2

3. Vector addition

u₁: 2

u₂: 1

v₁: 1

v₂: 3

The purple arrow is the total movement. The second vector starts at the tip of the first vector.

4. Scalar multiplication

base x: 3

base y: 2

scalar c: 1.5

5. Distance between two vectors

A x: 5

A y: 4

B x: 2

B y: 1

6. Feature vectors and scaling

Each restaurant is represented by [price level, distance, rating]. Raw distance can be dominated by the coordinate with the largest numerical scale.

A

price 2, distance 1.5, rating 4.6

B

price 3, distance 2.0, rating 4.8

C

price 1, distance 8.0, rating 4.2

Currently: raw features

Modeling warning: Linear algebra computes exactly what you ask it to compute. It does not automatically know which units are meaningful.

7. High-dimensional length simulation

Generate random vectors and watch how their lengths behave as dimension grows.

dimension: 50

number of vectors: 300

8. Reflection notebook

Use this space during class. Your text stays in your browser only.

Explain why a vector is more than a list of numbers. Give your own feature-vector example.

9. Chapter takeaway

A vector has algebraic rules, geometric pictures, and real-world meanings. Addition combines, scalar multiplication changes intensity, subtraction measures change, norm measures size, and distance compares objects.