The Third Dimension
3D Geometry, Lines, Planes & Spatial Proof
The Central Question
World 12 gave us (x,y). World 21 gave us vectors. Now add one more spatial direction: (x,y,z). Suddenly a point is no longer confined to a plane. We ask: What happens when vectors describe actual geometric objects in three-dimensional space?
2D PLANE →
3D COORDINATES →
POSITION VECTOR →
DIRECTION →
LINE →
PLANE →
ANGLE →
INTERSECTION →
DISTANCE →
SHORTEST PATH →
3D GEOMETRY
The 60 Spatial Cells
From 3D Coordinates to Intersecting Planes
CELL 01-08
3D Foundations
- 3D Coordinates (x,y,z)
- Origin & Three Axes
- Coordinate Planes & Octants
- 3D Distance Formula
- Position Vectors
CELL 09-13
Directions
- Vector Between Points
- Direction Ratios (a:b:c)
- Direction Cosines (l, m, n)
- l² + m² + n² = 1
- Angles with Axes
CELL 14-22
Lines in Space
- Vector Equation (r = a + λb)
- Cartesian Equation Form
- Parametric Form
- Line Through Two Points
- Parallel, Intersecting & Skew
CELL 23-25
Angles
- Angle Between Lines
- cos θ = |a·b| / (|a||b|)
- Spatial Orientation
- Projection
CELL 26-34
Planes
- Normal Vector (n)
- Vector Eq: (r - a)·n = 0
- Cartesian: Ax + By + Cz + D = 0
- Plane Through 3 Points (Cross Prod)
- Parallel & Intersecting Planes
CELL 35-40
Line-Plane Geometry
- Angle Between Planes
- Line-Plane Angle (sin θ)
- Line Parallel/Perpendicular to Plane
- Line-Plane Intersection
- Plane-Plane Intersection (Line)
CELL 41-45
Distance
- Point-Plane Distance
- Parallel-Plane Distance
- Shortest Distance (Skew Lines)
- Projection Geometry
CELL 46-51
3D Systems & Locus
- Coplanarity of Lines
- Systems of Planes
- 3D Locus Concept
- Sphere as a Distance Locus
- Plane as a Locus
CELL 52-59
Connections
- 2D → 3D Generalization
- Geometry ↔ Equation
- 3D Motion & Space Curves
- Computer Graphics & Robotics
- Physics Applications
CELL 60
The Third Dimension Synthesis
- Point → Vector → Line → Plane
- Dot Product → Angle
- Cross Prod → Normal / Area
- Triple Prod → Volume / Coplanar
Exit Portal
Once we can describe points, vectors, lines, and planes in space, how do we systematically transform entire spaces and solve enormous collections of linear relationships? That is the natural gateway into the grid of linear algebra.