3D Coordinate Basics

🚀 3D Coordinate Geometry: Your Map to the Universe!

The Big Picture

Imagine you’re a tiny ant living on a flat piece of paper. You can only move left-right and up-down. That’s 2D!

Now imagine you’re a bird. You can fly left-right, up-down, AND forward-backward. That’s 3D! Welcome to the third dimension!

🌍 The 3D Coordinate System

The Story

Think of your room as a giant 3D map. Pick one corner where the floor meets two walls. That corner is your origin — the starting point where everything begins.

From that corner:

• The X-axis goes along one wall (left-right)
• The Y-axis goes along the other wall (forward-backward)
• The Z-axis goes straight up to the ceiling (up-down)

Every point in your room can be described by three numbers: (x, y, z)

Simple Example

Your bed might be at position (3, 2, 0) — meaning:

• 3 steps along the first wall
• 2 steps along the second wall
• 0 steps up (it’s on the floor!)

A spider on your ceiling lamp might be at (3, 2, 8) — same x and y, but 8 feet up!

graph TD
    A["Origin 0,0,0"] --> B["X-axis: Left-Right"]
    A --> C["Y-axis: Forward-Backward"]
    A --> D["Z-axis: Up-Down"]
    B --> E["Point 5,0,0"]
    C --> F["Point 0,5,0"]
    D --> G["Point 0,0,5"]

The Golden Rule

Every point in 3D space needs exactly THREE numbers (x, y, z) to find it.

📏 Distance in 3D

The Story

Remember how you find distance on a flat map? You use the good old Pythagorean theorem!

In 3D, it’s almost the same — we just add one more dimension.

The Formula

Distance between points (x₁, y₁, z₁) and (x₂, y₂, z₂):

d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Why Does This Work?

Imagine a box. The distance between two corners is like going through the box diagonally. We combine:

1. How far apart horizontally (x)
2. How far apart depth-wise (y)
3. How far apart vertically (z)

Simple Example

Find the distance from A(1, 2, 3) to B(4, 6, 3)

Step 1: Find the differences

• x: 4 - 1 = 3
• y: 6 - 2 = 4
• z: 3 - 3 = 0

Step 2: Square them

• 3² = 9
• 4² = 16
• 0² = 0

Step 3: Add and take square root

• √(9 + 16 + 0) = √25 = 5 units

Pro Tip: When z-values are the same, it’s like measuring distance on a flat floor!

🎯 Midpoint Formula in 3D

The Story

You and your friend are standing at different spots in a park. Where should you meet so both of you walk the same distance?

The exact middle!

The Formula

Midpoint M of points (x₁, y₁, z₁) and (x₂, y₂, z₂):

M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

It’s just averaging each coordinate!

Simple Example

Find the midpoint of A(2, 4, 6) and B(8, 10, 12)

• x: (2 + 8) / 2 = 5
• y: (4 + 10) / 2 = 7
• z: (6 + 12) / 2 = 9

Midpoint M = (5, 7, 9) ✓

graph LR
    A["A#40;2,4,6#41;"] --> M["M#40;5,7,9#41;"]
    M --> B["B#40;8,10,12#41;"]

Why It Works

Think of it like finding the balance point of a seesaw. The middle is where both sides are equal!

✂️ Section Formula in 3D

The Story

What if you don’t want the exact middle? What if you want to divide a line in a specific ratio — like cutting a rope so one piece is twice as long as the other?

The Formula

Point P dividing line AB in ratio m:n

If A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂):

P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))

Memory Trick

• m is the part near A
• n is the part near B
• Cross-multiply: m goes with B’s coordinates, n goes with A’s coordinates

Simple Example

Find the point dividing A(1, 2, 3) and B(4, 5, 6) in ratio 2:1

Here m = 2, n = 1

• x: (2×4 + 1×1) / (2+1) = (8+1) / 3 = 3
• y: (2×5 + 1×2) / (2+1) = (10+2) / 3 = 4
• z: (2×6 + 1×3) / (2+1) = (12+3) / 3 = 5

Point P = (3, 4, 5) ✓

Special Case: External Division

If the point is outside the line segment, use ratio m:-n (negative!)

🧭 Direction of a Line in 3D

The Story

When giving directions, you don’t just say “go 5 kilometers.” You say “go 5 km north” or “go 5 km towards the mountain.”

In 3D, we describe direction using direction ratios or direction cosines.

Direction Ratios

If a line goes from A(x₁, y₁, z₁) to B(x₂, y₂, z₂), its direction ratios are:

a = x₂ - x₁
b = y₂ - y₁
c = z₂ - z₁

These tell you: “For every unit you move, go a in x, b in y, and c in z.”

Simple Example

Find direction ratios of line from A(1, 2, 3) to B(4, 6, 7)

• a = 4 - 1 = 3
• b = 6 - 2 = 4
• c = 7 - 3 = 4

Direction ratios = 3 : 4 : 4 (or simply written as (3, 4, 4))

Direction Cosines

These are the angles the line makes with each axis.

If l, m, n are direction cosines:

l = a/√(a²+b²+c²)
m = b/√(a²+b²+c²)
n = c/√(a²+b²+c²)

And the magic rule: l² + m² + n² = 1 (always!)

Example Continued

From direction ratios (3, 4, 4):

Length = √(3² + 4² + 4²) = √(9 + 16 + 16) = √41

• l = 3/√41
• m = 4/√41
• n = 4/√41

Why Direction Matters

Two lines are parallel if they have the same direction ratios (or proportional ones like 3:4:4 and 6:8:8).

Two lines are perpendicular if: a₁a₂ + b₁b₂ + c₁c₂ = 0

🎮 Quick Reference Table

🌟 The Journey So Far

You’ve just learned to:

1. Locate any point in 3D space with (x, y, z)
2. Measure the distance between any two points
3. Find the midpoint of any line segment
4. Divide a line in any ratio you want
5. Describe which direction a line is pointing

You now have the coordinates to navigate the 3D universe! Whether you’re designing video games, building rockets, or just understanding how GPS works — these are the tools that make it all possible.

You’re ready for the 3D world! 🚀