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➡️ Numbers as Arrows

How to Think About Vectors​

🎯 The Simple Story​

Imagine you're on a treasure hunt!

The treasure map shows you directions:

"Start at the big oak tree, take 5 steps East, then 3 steps North"

You're not just standing at a point - you're following an arrow!

That arrow = that's a vector!

Vectors tell you: "Go this far, in this direction."

🧠 Mental Model​

Hold this picture in your head:

A vector is like an arrow from start to finish

    End point
       ↑
       │ 3 steps North
       │
       •────→ 5 steps East
     Start

    We write this as: (5, 3)

Different vectors on the same map:

Vector A: "6 East" → (6, 0)
Vector B: "2 North" → (0, 2)  
Vector C: "5 East, 3 North" → (5, 3)
Vector D: "2 West, 4 South" → (-2, -4)

📊 See It Happen​

Let's watch someone following vectors:

🎮 Try It Yourself​

Question 1: You start at the big oak tree. The map says: "4 steps West, 2 steps South." Where are you?

Show Answer

Starting point: (0, 0)

4 steps West = -4 (going opposite direction of East) 2 steps South = -2 (going opposite direction of North)

Final position: (-4, -2)

Think of it as coordinates on graph paper, not just arrows!

Question 2: You're at position (3, 3). The map says "2 steps East, 1 step South." Where do you end up?

Show Answer

Starting: (3, 3)

2 steps East: add 2 to the first number = 5 1 step South: subtract 1 from the second number = 2

Final position: (5, 2)

Question 3: Which arrow is the vector (4, -1)?

Show Answer
         North (positive y)
              ↑
              │
              │
              │
              │
West (negative x)<────✓────> East (positive x)
              │    (4, -1) means:
              │    - 4 East = to the right
              │    - -1 South = down 1
              │
              ↓
         South (negative y)

So (4, -1) = "4 steps East, 1 step South"

🔢 The Math​

Vector Notation​

Mathematicians write vectors in two ways:

Arrow notation (what we use): v=(vx,vy)\mathbf{v} = (v_x, v_y)

Where:

  • vxv_x is the horizontal component (east/west)
  • vyv_y is the vertical component (north/south)

Example: v=(5,3)\mathbf{v} = (5, 3) means 5 east, 3 north.

Column notation (you'll see sometimes): [53]\begin{bmatrix} 5 \\ 3 \end{bmatrix}

Same thing, just written differently!

Two Dimensions vs Many Dimensions​

So far we've used 2D (two-dimensional) vectors:

Map: Two directions only
- East/West
- North/South

Vector: (x, y)

But vectors can have more dimensions:

3D (three dimensions):
  - East/West
  - North/South
  - Up/Down (flying)
  
  Vector: (x, y, z)

256D (256 dimensions!):
  - Imagine 256 different "directions"
  - ML-KEM uses this for polynomials!
  
  Vector: (a_0, a_1, a_2, ..., a_255)

ML-KEM's vectors have 256 "steps" in 256 different directions!

💡 Why We Care​

Problem: ML-KEM Doesn't Work With Single Numbers​

If ML-KEM only used single numbers like "5" or "17", it would be weak. Everyone could guess them!

Solution: Use vectors with many coordinates!

Single number secret:
  - Secret: 42
  - Eve tries: 41, 42, 43... Easy!

Vector secret (3D):
  - Secret: (7, 2, 5)
  - Eve tries: (7, 2, 5), (7, 2, 6), (7, 3, 5)...
  - Much harder! More possibilities!

256D vector (ML-KEM):
  - Secret: (a_0, a_1, a_2, ..., a_255)
  - Eve would need to try: huge number of combinations!
  - Practically impossible!

Vectors make secrets exponentially harder to guess!

✅ Quick Check​

Can you explain this to a 5-year-old?

Try saying this out loud:

"Imagine you have a treasure map with arrows. Each arrow tells you: 'Go this far in this direction.' A vector is just one of those arrows - it tells you how many steps to take and in which directions."

Can you draw a picture?

Draw this:
  1. Draw a piece of paper (graph paper is best)
  2. Draw a dot in the center (start)
  3. Draw an arrow pointing up-right: label it "5 steps right, 3 steps up"
  4. Label the endpoint: (5, 3)
  5. Draw another arrow going left-down: label it "2 steps left, 4 steps down"
  6. Label the endpoint: (-2, -4)

🎓 Key Takeaways​

✅ Vector = Arrow from start to end ✅ Notation = (x, y) where x is horizontal, y is vertical ✅ Coordinates = Where you end up after following the arrow ✅ Multiple dimensions = Can have 2, 3, 256+ directions ✅ ML-KEM = Uses 256-dimensional vectors (hard for attackers!) ✅ Security benefit = More dimensions = exponentially harder to guess

🎉 What You'll Learn Next​

Now you understand vectors! Next, we'll learn about:

How Arrows Add Up → Vector Operations

What happens when you follow multiple arrows in a row!