Lines and Planes in 3D

🚀 3D Coordinate Geometry: Lines and Planes in 3D

The Story of 3D Space

Imagine you’re a tiny ant walking on a piece of paper. You can only go forward-backward and left-right. That’s 2D!

Now imagine you’re a bird. You can go forward-backward, left-right, AND up-down. That’s 3D! 🐦

In 3D, we use three numbers (x, y, z) to describe any point in space. Think of it like giving directions in a giant invisible cube.

📍 3D Line Equation Forms

What is a 3D Line?

A line in 3D is like a laser beam shooting through space. It goes on forever in both directions.

To describe this laser beam, we need:

1. A starting point (where we first see the laser)
2. A direction (which way the laser points)

Three Ways to Write a 3D Line

1. Vector Form (The Easy Way)

Formula: r = a + λb

• a = starting point (a position vector)
• b = direction the line travels
• λ = how far along the line you walk

Example: A line through point (1, 2, 3) going in direction (2, 1, -1)

r = (1, 2, 3) + λ(2, 1, -1)

Think of it like: “Start at the coffee shop, walk toward the park.”

2. Cartesian Form (The Fraction Way)

Formula: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c

Example: Line through (1, 2, 3) with direction (2, 1, -1)

(x - 1)/2 = (y - 2)/1 = (z - 3)/(-1)

3. Parametric Form (The Recipe Way)

Formula:

• x = x₁ + aλ
• y = y₁ + bλ
• z = z₁ + cλ

Example:

x = 1 + 2λ
y = 2 + λ
z = 3 - λ

Each value of λ gives you a different point on the line!

graph TD
    A["3D Line"] --> B["Vector Form"]
    A --> C["Cartesian Form"]
    A --> D["Parametric Form"]
    B --> E["r = a + λb"]
    C --> F["#40;x-x₁#41;/a = #40;y-y₁#41;/b = #40;z-z₁#41;/c"]
    D --> G["x=x₁+aλ, y=y₁+bλ, z=z₁+cλ"]

📐 Angle Between Lines in 3D

The Story

Two roads meet at a crossroads. The angle between them tells you how sharp the turn is!

The Formula

If line 1 has direction b₁ = (a₁, b₁, c₁) and line 2 has direction b₂ = (a₂, b₂, c₂):

cos θ = |b₁ · b₂| / (|b₁| × |b₂|)

Breaking It Down

1. Dot product: b₁ · b₂ = a₁a₂ + b₁b₂ + c₁c₂
2. Magnitude: |b₁| = √(a₁² + b₁² + c₁²)

Example

Line 1 direction: (1, 2, 2) Line 2 direction: (2, 1, -2)

Step 1: Dot product = (1×2) + (2×1) + (2×-2) = 2 + 2 - 4 = 0

Step 2: Since dot product = 0, the lines are perpendicular (90°)!

🎯 Quick Tip: If the dot product is 0, the lines meet at a right angle!

✂️ Skew Lines

What Are Skew Lines?

Imagine two pencils floating in the air:

• They don’t touch (not intersecting)
• They’re not parallel either
• They’re like two planes flying at different heights going different directions

These are SKEW LINES!

How to Identify Skew Lines

Two lines are skew if:

1. They are NOT parallel (different direction ratios)
2. They do NOT intersect (no common point)

Example

Line 1: (x-1)/2 = (y-2)/3 = (z-3)/4 Line 2: (x-2)/5 = (y-4)/2 = (z-6)/3

Check parallel: Directions are (2,3,4) and (5,2,3)

• 2/5 ≠ 3/2 → NOT parallel ✓

Check intersection: Solve for a common point

• If no solution exists → SKEW! ✓

graph TD
    A["Two Lines in 3D"] --> B{Same Direction?}
    B -->|Yes| C["Parallel Lines"]
    B -->|No| D{Do They Meet?}
    D -->|Yes| E["Intersecting Lines"]
    D -->|No| F["Skew Lines"]

📏 Distance Between Skew Lines

The Problem

How far apart are two flying pencils that never touch?

The Formula

For lines:

• Line 1: r = a₁ + λb₁
• Line 2: r = a₂ + μb₂

Distance = |((a₂ - a₁) · (b₁ × b₂))| / |b₁ × b₂|

Step-by-Step

1. Find a₂ - a₁ (vector between points on each line)
2. Find b₁ × b₂ (cross product of directions)
3. Find the dot product of results from steps 1 and 2
4. Divide by magnitude of step 2

Example

Line 1: Through (1,1,1) with direction (2,-1,1) Line 2: Through (2,1,-1) with direction (3,-5,2)

Step 1: a₂ - a₁ = (2-1, 1-1, -1-1) = (1, 0, -2)

Step 2: b₁ × b₂ = (2,-1,1) × (3,-5,2) = ((-1×2)-(1×-5), (1×3)-(2×2), (2×-5)-(-1×3)) = (3, -1, -7)

Step 3: (1,0,-2) · (3,-1,-7) = 3 + 0 + 14 = 17

Step 4: |b₁ × b₂| = √(9+1+49) = √59

Distance = 17/√59 ≈ 2.21 units

🪟 Equation of a Plane

What is a Plane?

A plane is like an infinite sheet of paper floating in space. It goes on forever!

Three Ways to Write a Plane Equation

1. General Form (The Classic)

ax + by + cz + d = 0

Where (a, b, c) is the normal vector (perpendicular to the plane).

Example: 2x + 3y + 4z - 5 = 0

2. Vector Form

r · n = d

Where n is the normal vector and d is the distance from origin.

Example: r · (2, 3, 4) = 5

3. Intercept Form

x/a + y/b + z/c = 1

Where a, b, c are the intercepts on the x, y, z axes.

Example: Plane cutting x-axis at 3, y-axis at 4, z-axis at 6:

x/3 + y/4 + z/6 = 1

Special Planes to Remember

⬆️ Normal to a Plane

What is a Normal?

The normal is a vector that stands straight up from the plane, like a flagpole on a field!

Finding the Normal

From equation ax + by + cz + d = 0:

• Normal vector = (a, b, c)

Example

Plane: 3x - 2y + 5z = 10 Normal: (3, -2, 5)

Why is the Normal Important?

• It tells us which way the plane “faces”
• It helps calculate angles
• It helps find distances

🌟 Fun Fact: Two planes are parallel if they have the same (or proportional) normal vectors!

📍 Distance from Point to Plane

The Story

You’re standing at a point. How far is the nearest part of the wall (plane)?

The Formula

Distance from point (x₁, y₁, z₁) to plane ax + by + cz + d = 0:

Distance = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)

Example

Point: (2, 3, 1) Plane: 2x + 2y + z - 9 = 0

Step 1: Plug into numerator |2(2) + 2(3) + 1(1) - 9| = |4 + 6 + 1 - 9| = |2| = 2

Step 2: Calculate denominator √(4 + 4 + 1) = √9 = 3

Distance = 2/3 units

graph TD
    A["Point P"] --> B["Drop Perpendicular"]
    B --> C["Plane"]
    B --> D["Distance = |ax₁+by₁+cz₁+d|/√#40;a²+b²+c²#41;"]

🔺 Angle Between Planes

The Story

Two walls meet at a corner. What’s the angle of that corner?

The Formula

For planes with normals n₁ = (a₁, b₁, c₁) and n₂ = (a₂, b₂, c₂):

cos θ = |n₁ · n₂| / (|n₁| × |n₂|)

Example

Plane 1: 2x + y - 2z = 5 → n₁ = (2, 1, -2) Plane 2: 3x - 6y - 2z = 7 → n₂ = (3, -6, -2)

Step 1: n₁ · n₂ = 6 - 6 + 4 = 4

Step 2: |n₁| = √(4+1+4) = 3 |n₂| = √(9+36+4) = 7

Step 3: cos θ = |4| / (3 × 7) = 4/21

θ = cos⁻¹(4/21) ≈ 79°

Special Cases

🎯 Line and Plane Intersection

Three Possible Scenarios

1. Line crosses plane → One intersection point
2. Line lies in plane → Infinite points (the whole line!)
3. Line is parallel to plane → No intersection

Finding the Intersection Point

Method:

1. Write line in parametric form
2. Substitute into plane equation
3. Solve for λ
4. Put λ back to get the point

Example

Line: (x-1)/2 = (y-2)/3 = (z-3)/4 = λ Plane: 2x + 3y - z = 4

Step 1: Parametric form

• x = 1 + 2λ
• y = 2 + 3λ
• z = 3 + 4λ

Step 2: Substitute into plane 2(1+2λ) + 3(2+3λ) - (3+4λ) = 4 2 + 4λ + 6 + 9λ - 3 - 4λ = 4 5 + 9λ = 4 9λ = -1 λ = -1/9

Step 3: Find the point

• x = 1 + 2(-1/9) = 7/9
• y = 2 + 3(-1/9) = 15/9 = 5/3
• z = 3 + 4(-1/9) = 23/9

Intersection Point: (7/9, 5/3, 23/9)

graph TD
    A["Line meets Plane"] --> B{Check λ}
    B -->|One value| C["One Point"]
    B -->|All values work| D["Line in Plane"]
    B -->|No value works| E["No Intersection"]

🎓 Quick Summary

🌟 You Did It!

You’ve just learned how to:

• ✅ Write lines in 3D using three different forms
• ✅ Find angles between lines
• ✅ Identify and measure skew lines
• ✅ Write plane equations
• ✅ Work with normals
• ✅ Calculate distances
• ✅ Find where lines meet planes

3D geometry is like having superpowers to navigate any space in the universe! 🚀