3. Vectors in 2D and 3D: Visualization and Coordinates

Capsule overview

In this capsule you're going to learn to visualize and work with vectors in two and three dimensions. We move from the abstract definition (the previous capsule) to concrete visualization: how to draw vectors on a Cartesian plane, how to read coordinates, how to interpret directions.

This is the most visual capsule in the module. If in capsule 02 you understood WHAT a vector is (direction + magnitude), here you'll see HOW it looks and how to work with it in spaces you can actually draw. This geometric intuition is the foundation for understanding high-dimensional spaces (Capsule 04) and for everything that comes after.

When you finish, you'll be able to draw 2D and 3D vectors on paper, read coordinates without confusion, and identify directions (northeast, southwest, up-right, etc.). That visualization skill is what will let you understand embeddings and semantic search: even though real embeddings are 1536D (not drawable), the intuition of "nearby points = similar meanings" comes from being able to visualize 2D and 3D.

The Cartesian plane (2D)

The Cartesian plane is the 2D space where we place vectors. It has two perpendicular axes:

      Y ↑ (vertical)
        |
      3 |
        |
      2 |
        |
      1 |
        |
      0 |─────────────→ X (horizontal)
        0  1  2  3  4  5

Components:

• X axis (horizontal): Positive to the right, negative to the left
• Y axis (vertical): Positive up, negative down
• Origin: The point (0, 0) where the axes cross

Any point on the plane can be identified with coordinates (x, y):

• (3, 2) = 3 units to the right, 2 up
• (-2, 1) = 2 units to the left, 1 up
• (0, -3) = On the Y axis, 3 units down

Analogy: Think of the Cartesian plane as a grid of streets. The X axis is "how many blocks east or west" and the Y axis is "how many blocks north or south". The origin is your house (the reference point). Any location is described with two numbers: (east-west, north-south).

Drawing vectors in 2D

To draw a vector [x, y]:

Step 1: Mark the origin

      ↑ Y
      |
    0 •───────→ X
      0

The origin is (0, 0).

Step 2: Locate the destination point

For the vector [3, 2]:

• Move 3 units along X (right)
• Move 2 units along Y (up)
• Mark the point (3, 2)

      ↑ Y
      |
    2 |       •(3,2) ← Destination point
      |
    1 |
      |
    0 •───────────→ X
      0  1  2  3

Step 3: Draw the arrow

Trace an arrow from the origin (0, 0) to the point (3, 2):

      ↑ Y
      |
    2 |       •(3,2)
      |      ╱
    1 |     ╱
      |    ╱ ← Arrow (vector)
    0 •───────────→ X
      0  1  2  3

That is the vector [3, 2]: An arrow from the origin to (3, 2).

Examples of 2D vectors

Example 1: Vector [4, 1]

      ↑ Y
      |
    1 |           •(4,1)
      |          ╱
    0 •─────────────────→ X
      0  1  2  3  4

Direction: East-northeast (mostly horizontal)
Magnitude: √(4² + 1²) = √17 ≈ 4.12

Example 2: Vector [-2, 3]

      ↑ Y
      |
    3 |   •(-2,3)
      |  ╱
    2 | ╱
      |╱
    1 |
      |
    0 •───────────→ X
     -2 -1  0  1

Direction: Northwest (negative x, positive y)
Magnitude: √((-2)² + 3²) = √13 ≈ 3.61

Example 3: Vector [0, -3]

      ↑ Y
      |
    0 •───────────→ X
      |
   -1 |
      |
   -2 |
      |
   -3 •(0,-3)

Direction: Due south (straight down)
Magnitude: 3 (only on the Y axis)

Example 4: Vector [-4, -2]

      ↑ Y
      |
    0 •───────────→ X
      | ╲
   -1 |  ╲
      |   ╲
   -2 |    •(-4,-2)
     -4 -3 -2 -1  0

Direction: Southwest (both negative)
Magnitude: √(16 + 4) = √20 ≈ 4.47

The four quadrants (2D)

On the Cartesian plane there are four quadrants according to the signs of x and y:

        Y ↑
          |
    II    |    I
  (-, +)  |  (+, +)
          |
  ────────•────────→ X
          |
   III    |   IV
  (-, -)  |  (+, -)
          |

Quadrant I: x > 0, y > 0 → Northeast
Quadrant II: x < 0, y > 0 → Northwest
Quadrant III: x < 0, y < 0 → Southwest
Quadrant IV: x > 0, y < 0 → Southeast

Examples:

• [3, 2] → Quadrant I (northeast)
• [-2, 3] → Quadrant II (northwest)
• [-4, -2] → Quadrant III (southwest)
• [4, -1] → Quadrant IV (southeast)

Why it matters: When you visualize embeddings in 2D (dimensionality reduction with t-SNE or UMAP, which you'll see later), the points can be in any quadrant. Understanding quadrants helps you interpret directions and clusters.

3D space (three dimensions)

In 3D we add a third axis: Z (depth).

Coordinate system:

        Y ↑ (vertical)
          |
          |
          •───────→ X (horizontal right)
         ╱
        ╱
       ↙ Z (depth toward you)

3D vector: [x, y, z]

Example: [3, 2, -1]

• x = 3 (right)
• y = 2 (up)
• z = -1 (backward / into the paper)

Visualization (2D projection):

      Y ↑
        |
      2 |       •(3,2,-1)
        |      ╱|
      1 |     ╱ |
        |    ╱  |
      0 •───────────→ X
        0  1  2  3
       (z=-1 is "inside" the paper)

Interpretation: If you're at the origin and you walk 3 steps east, 2 north, and 1 downward (or backward), you arrive at the point (3, 2, -1).

Examples of 3D vectors

Example 1: Vector [2, 3, 1]

Coordinates: x=2, y=3, z=1

Interpretation:

• 2 units along X (right)
• 3 units along Y (up)
• 1 unit along Z (toward you / out of the paper)

Magnitude: √(2² + 3² + 1²) = √(4 + 9 + 1) = √14 ≈ 3.74

Example 2: Vector [0, 0, 5]

Coordinates: x=0, y=0, z=5

Interpretation:

• Only on the Z axis (pure outward / depth)
• No movement in X or Y

Magnitude: 5 (only on one axis)

Visualization: An arrow coming out perpendicular from the XY plane toward you.

Example 3: Vector [-2, 1, -3]

Coordinates: x=-2, y=1, z=-3

Interpretation:

• 2 units to the left (negative x)
• 1 unit up (positive y)
• 3 units backward / inward (negative z)

Direction: Northwest-back (a combination of three components)

Magnitude: √(4 + 1 + 9) = √14 ≈ 3.74

3D visualization: The challenge

Problem: It's hard to draw 3D on paper (which is 2D).

Solutions:

1. Isometric projection

You draw the 3 axes at angles:

        Y ↑
          |
          |
          •───────→ X
         ╱
        ╱
       ↙ Z

It's a visual convention that works on paper.

2. Use physical objects

A 3D vector is like the position of an object in a room:

• X = how far right/left
• Y = how far up/down (height)
• Z = how far forward/back (depth)

Example: A glass on your desk has a position (x, y, z):

• x = horizontal position on the table
• y = height from the floor
• z = depth (near or far from you)

That's a 3D "vector": its location relative to an origin (e.g. the corner of the room).

3. Digital tools (optional)

Tools like GeoGebra 3D, Desmos 3D or Python with Matplotlib let you rotate and manipulate 3D vectors interactively. It isn't necessary for this module, but if you want to experiment, they're freely available.

Comparison: 2D vs 3D

Key similarity: The concept is the same. A 3D vector is simply a 2D vector with one additional axis. The intuition of "direction + magnitude" holds.

Quadrants vs octants

In 2D there are 4 quadrants (according to the signs of x, y).

In 3D there are 8 octants (according to the signs of x, y, z):

You don't need to memorize the 8 octants. Just understand that in 3D there are more "regions" of space than in 2D, and that the coordinates (positive/negative signs) determine which region you're in.

Why it matters: When you visualize embeddings reduced to 3D, the points will be spread across the 8 octants. Understanding that all regions are "valid" (there are no "forbidden" regions) helps you interpret visualizations.

How to read coordinates without confusion

Basic rule: Order matters

In a vector, the coordinates are in order:

2D: [x, y] — always X first, Y second
3D: [x, y, z] — always X first, Y second, Z third

Examples:

• [3, 2] ≠ [2, 3] (different points)
• [3, 2, 1] ≠ [1, 2, 3] (different points)

Technique: "Read axis by axis"

For the vector [4, -2, 3]:

1. X axis (first): 4 → Move 4 units to the right
2. Y axis (second): -2 → Move 2 units down
3. Z axis (third): 3 → Move 3 units outward (depth)

Result: You arrive at the point (4, -2, 3).

Guided practice:

Read these coordinates axis by axis:

a) [5, 0]

• X: 5 (right)
• Y: 0 (you don't move vertically)
• Result: A point on the pure X axis

b) [0, -3]

• X: 0 (you don't move horizontally)
• Y: -3 (3 units down)
• Result: A point on the negative Y axis

c) [2, 2, 2]

• X: 2 (right)
• Y: 2 (up)
• Z: 2 (outward)
• Result: A point in octant 1 (all positive)

Directions in 2D: The compass rose

To describe the directions of 2D vectors, you can use map terminology:

           N (North)
           ↑ Y
           |
  NW ╱─────•─────╲ NE
    ╱      |      ╲
   ╱       |       ╲
W ←────────•────────→ E (East)
   ╲       |       ╱
    ╲      |      ╱
  SW ╲─────•─────╱ SE
           |
           ↓
           S (South)
         ─→ X

Examples:

• [3, 2] → NE (northeast)
• [-2, 3] → NW (northwest)
• [4, 0] → E (due east)
• [0, -3] → S (due south)
• [-1, -1] → SW (southwest)

Practical use: When you describe vector directions, you can say "the vector points northeast" instead of "the vector has positive x and positive y". It's more intuitive.

Directions in 3D: Adding depth

In 3D, we add up/down (if Z is height) or forward/back (if Z is depth):

Example 1: [2, 3, 1]

• Direction: Northeast-up (x+, y+, z+)

Example 2: [-2, 1, -3]

• Direction: Northwest-back (x-, y+, z-)

Example 3: [0, 0, 5]

• Direction: Straight up / straight forward (only z+)

There are no standard names like "northeast" for 3D; you simply describe the axes: "up-right-forward" or "left-down-back".

Magnitude in 2D and 3D

The magnitude (the length of the vector) is calculated with the extended Pythagorean theorem:

In 2D:

Vector [x, y]
Magnitude = √(x² + y²)

Example: [3, 4] → Magnitude = √(9 + 16) = √25 = 5

In 3D:

Vector [x, y, z]
Magnitude = √(x² + y² + z²)

Example: [2, 3, 6] → Magnitude = √(4 + 9 + 36) = √49 = 7

You don't need to calculate exact magnitudes in every exercise. What matters is understanding that:

• Magnitude = the distance from the origin to the point
• Magnitude is always ≥ 0 (it can't be negative)
• Magnitude = 0 only for the zero vector [0, 0] or [0, 0, 0]

Special vectors: The unit axes

The unit vectors are vectors with magnitude 1 that point in the direction of each axis:

In 2D:

Unit X vector: [1, 0]  (points east, magnitude = 1)
Unit Y vector: [0, 1]  (points north, magnitude = 1)

Visualization:

      ↑ Y
      |
    1 •(0,1) ← Unit Y
      |
    0 •───•─────→ X
      0   (1,0)
          ↑ Unit X

In 3D:

Unit X vector: [1, 0, 0]
Unit Y vector: [0, 1, 0]
Unit Z vector: [0, 0, 1]

Why it matters: Any vector can be decomposed as a combination of unit vectors:

[3, 2] = 3 × [1, 0] + 2 × [0, 1]
[2, 3, 1] = 2 × [1, 0, 0] + 3 × [0, 1, 0] + 1 × [0, 0, 1]

That idea of "decomposing into axes" is fundamental in linear algebra, but you don't need to memorize it. Just understand that the axes are the "pure directions" (north, east, up) and that any direction is a combination of them.

Useful analogies for 3D visualization

Analogy 1: A room

Imagine you're in the corner of a room (the origin):

• X: Left-right (the width of the room)
• Y: Down-up (height)
• Z: Forward-back (depth)

An object at [2, 1, 3] is:

• 2 meters to the right
• 1 meter above the floor
• 3 meters toward the center of the room

Analogy 2: A drone in the air

A flying drone has a 3D position:

• X, Y: Position on the map (horizontal)
• Z: Height above the ground

Its position [10, 5, 20] means:

• 10m east, 5m north, 20m of altitude

Analogy 3: Video games

In 3D games, every character or object has a position [x, y, z]. When you move, you're changing your position vector in 3D space.

Connection to embeddings and AI

Now that you can visualize 2D and 3D vectors, understand this:

In AI, words are represented as HIGH-dimensional vectors (300D, 768D, 1536D, etc.).

Conceptual example:

If we represented 3 words in 2D (simplified):

      Y ↑
        |
      2 |    •"cat"
        |   ╱
      1 | •"dog"
        |
      0 |─────────────→ X
        |         •"car"
        0  1  2  3  4  5

Observation:

• "dog" and "cat" are close together (both animals, pets)
• "car" is far away (a different concept)

That is semantic search: representing concepts as vectors, such that similar concepts → nearby vectors.

In reality:

• They aren't 2D but 1536D or more
• We can't draw them
• But the intuition is the same: geometric closeness = semantic similarity

We'll look at this in depth in Capsule 06 (Why vectors in AI) and in Module 2 (Vector Spaces).

Exercises

Exercise 1: Draw 2D vectors

Draw on paper (or mentally) these vectors:

a) [5, 2]
b) [-3, 4]
c) [2, -3]
d) [0, 4]

For each one:

• Mark the origin
• Locate the point
• Draw the arrow
• Identify the quadrant and direction

      ↑ Y
    2 |         •(5,2)
      |        ╱
    0 •───────────────→ X
      0    3    5

      ↑ Y
    4 |   •(-3,4)
      |  ╱
    2 | ╱
      |╱
    0 •───────────→ X
     -3  -1  0

      ↑ Y
      |
    0 •───────────→ X
      | ╲
   -2 |  ╲
      |   •(2,-3)
      0  1  2

      ↑ Y
    4 •(0,4)
      |
    2 |
      |
    0 •───────────→ X
      0

Exercise 2: Identify coordinates from a diagram

Look at this diagram and write the coordinates of each point:

      ↑ Y
      |
    3 | A•      •B
      |
    1 |     •C
      |
    0 •───────────────→ X
   -2 |  •D
     -2  0  2  4  6

Write the vectors (coordinates) of A, B, C, D.

Exercise 3: Estimating magnitude

Without calculating exactly, order these vectors from smallest to largest magnitude:

a) [1, 1]
b) [3, 0]
c) [2, 2]
d) [0, 5]

Exercise 4: 3D vectors in the room

Imagine you're in the corner of a room (the origin). Describe the position of these points in natural language:

a) [3, 2, 0]
b) [0, 0, 5]
c) [2, 1, 3]

Exercise 5: Opposite vectors

What is the opposite vector (same magnitude, opposite direction) of each one?

a) [3, 2]
b) [-1, 4]
c) [2, -3, 1]

Exercise 6: Mental 3D visualization

Without drawing, mentally describe the position of the vector [4, 0, -2]:

• Which octant is it in?
• Where does it point (in words)?
• Is it closer to the XY plane or the XZ plane?

Comparison: Different ways to represent the same vector

A vector can be expressed in several equivalent forms:

Form 1: Coordinates (standard)

v = [3, 2]

Form 2: As a point

Point P at (3, 2)

Form 3: As a displacement

"Move 3 units along X, 2 units along Y"

Form 4: As an arrow

Arrow from (0,0) to (3,2)

They're all equivalent. Use whichever is most useful for the context:

• In programming: [3, 2] (coordinates)
• In geometry: A visual arrow
• In physics: A displacement
• In AI: A point in space (an embedding)

Troubleshooting

Problem 1: "I can't visualize 3D on paper"

Symptom: You draw 2D vectors fine, but 3D confuses you.

Solution: Use physical objects. A glass on your desk has a 3D position:

• X = horizontal position (left-right on the table)
• Y = height (from the floor)
• Z = depth (near-far from you)

Move the glass mentally and describe its new position as [x, y, z]. That's more intuitive than drawing.

Alternatively: Accept that you can't draw 3D perfectly on paper and use isometric projections (diagrams with 3 axes like we did above). You don't need perfection; just intuition.

Problem 2: "I confuse [3, 2] with [2, 3]"

Symptom: You mix up the order of coordinates.

Solution: Order matters. It's always [X, Y] in 2D, [X, Y, Z] in 3D.

Mnemonic: "X comes before Y in the alphabet" → X is the first coordinate.

Quick check:

• [3, 2] → 3 along X (horizontal), 2 along Y (vertical)
• [2, 3] → 2 along X (horizontal), 3 along Y (vertical)

They're different points:

      ↑ Y
      |
    3 |    •[2,3]
      |
    2 |       •[3,2]
      |
    0 •───────────────→ X
      0  1  2  3

Problem 3: "Why is the origin (0, 0)?"

Symptom: You don't understand why all vectors "start from the origin".

Solution: It's a convention. In geometry, the origin is the reference point (like your house on a map). All vectors are measured from there so you have a consistent system.

If you want to represent "from A to B": You calculate B - A (vector subtraction, which you'll see in Capsule 05) and you get a vector from the origin that represents that displacement.

Example: To go from (1, 1) to (4, 3):

• Displacement: [4-1, 3-1] = [3, 2]
• You represent it as the vector [3, 2] from the origin

Don't worry if this doesn't fully click yet; Capsule 05 (Operations) will clear it up.

Problem 4: "I don't understand how [3, 2, -1] can be 'back'"

Symptom: The negative Z coordinate confuses you.

Solution: Negative Z = the opposite direction on the Z axis. If positive Z is "toward you" (out of the paper), negative Z is "backward" (into the paper). If positive Z is "up", negative Z is "down".

Convention: In 3D graphics, Z is sometimes height (Z+ = up, Z- = down). Other times Z is depth (Z+ = toward you, Z- = away). There's no single standard; it depends on the context. In this guide we use Z as depth (Z+ = toward you).

Example: [0, 0, -5] means "5 units backward" (into the paper or the screen).

Advanced visualization: Dimensionality reduction

When in later modules or in Guide #6 you see visualizations of embeddings (e.g. t-SNE, UMAP), you'll be looking at 2D or 3D projections of high-dimensional vectors (1536D).

How it works (conceptually):

Original vector (1536D):
[0.23, -0.45, 0.12, ..., -0.34]  (1536 numbers)

↓ Dimensionality reduction (t-SNE or UMAP)

Reduced vector (2D):
[2.3, 1.8]  (only 2 numbers)

The result: A point in 2D that you can draw, which "approximates" the position of the original vector in the high-dimensional space.

Why it matters: The embedding visualizations you see in blogs or papers are 2D/3D projections. Now that you know what a 2D/3D vector is, you'll be able to interpret those visualizations: nearby points = similar concepts (in the original high-dimensional space).

We'll look at dimensionality reduction conceptually in Module 2 (Vector Spaces). For now, just understand that a high-dimensional vector can be "projected" into 2D/3D to visualize it.

Summary

In one sentence: 2D vectors have 2 coordinates [x, y] and are drawn as arrows on a plane; 3D vectors have 3 coordinates [x, y, z] and are visualized in space (or with projections); both have direction and magnitude.

Key points:

• Cartesian plane: X (horizontal) and Y (vertical) axes in 2D; add the Z axis (depth) for 3D
• Drawing vectors: Origin → destination point → arrow
• Quadrants (2D): 4 regions according to the signs of x, y
• Octants (3D): 8 regions according to the signs of x, y, z
• Direction: North, south, east, west in 2D; add up/down or forward/back in 3D
• Magnitude: The distance from the origin (Pythagoras in 2D/3D)
• Unit vectors: [1, 0], [0, 1] in 2D; [1, 0, 0], [0, 1, 0], [0, 0, 1] in 3D
• Order matters: [3, 2] ≠ [2, 3]; always [X, Y] or [X, Y, Z]
• Embeddings: High-dimensional vectors (1536D); the same intuition as 2D/3D but not drawable
• Visualization: 2D/3D projections of embeddings use the same geometry you saw here

Connection to the next capsule

In Capsule 04 (Vectors in high dimensions), we'll extend what you saw here (2D, 3D) to spaces with 100D, 1536D, 3072D. You won't be able to draw them, but the geometric intuition (direction, magnitude, closeness) holds. If the 2D/3D visualization from this capsule isn't clear to you, review it before continuing; Capsule 04 assumes you can mentally visualize 2D and 3D vectors.

Additional resources

1. GeoGebra 3D Calculator — A free online tool for drawing and rotating 3D vectors interactively. You can enter [3, 2, 1] and see the vector in 3D. Very useful for experimenting.

2. Desmos 3D Calculator (Beta) — Another visual tool for 3D vectors. Similar to GeoGebra but with a simpler interface.

3. 3Blue1Brown: "Vectors" (Chapter 1) — A 10-minute video that visualizes vectors as arrows and as points. Perfect for reinforcing this capsule.

4. Khan Academy: "Position vectors" — A video on vectors as positions. It complements the idea of "vector as a point".

5. Better Explained: "An Intuitive Guide to Linear Algebra" — A blog with intuitive explanations of vectors and visualization. The "Vectors" section is useful.

6. Immersive Math: Chapter 2 - Vectors — An interactive book where you can manipulate 2D/3D vectors with the mouse. Excellent for geometric intuition.

Next capsule: 04-vectors-in-high-dimensions.md — What happens in 100D, 1536D, 3072D? A conceptual extension from 2D/3D to non-visualizable spaces.