The Vector World
Magnitude, Direction & The Geometry of Action
The Central Question
Coordinates describe where something is. Vectors describe how something points, moves, or acts. A scalar tells us how much. A vector tells us how much—and in which direction. Vectors bridge coordinate geometry, mechanics, multidimensional space, and linear algebra.
NUMBER →
MAGNITUDE →
DIRECTION →
VECTOR →
COMPONENTS →
VECTOR OPERATIONS →
DOT PRODUCT →
CROSS PRODUCT →
3D SPACE →
PHYSICAL SYSTEM
The 67 Vector Cells
From Arrows to Multidimensional Physics
CELL 01-11
Foundations
- Scalar vs Vector
- Vector as an Arrow
- Notation & Magnitude
- Unit Vectors (i, j, k)
- Zero, Negative & Parallel Vectors
CELL 12-19
Coordinate Vectors
- Position Vector
- Vector Between Two Points
- Head-to-Tail & Parallelogram Law
- Addition & Subtraction
- Scalar Multiplication
CELL 20-22
Components
- Component Resolution (Vx, Vy)
- Resultant from Components
- Vector Reconstruction
CELL 23-29
The Dot Product
- Definition & Component Form
- Test for Perpendicularity (= 0)
- Angle Between Two Vectors
- Vector Projection
- Work Done (W = F · d)
CELL 30-37
The Cross Product
- Direction & Right-Hand Rule
- Component Form (Determinant)
- Parallel Test (= 0)
- Area of Parallelogram & Triangle
- Torque & Angular Momentum
CELL 38-40
Advanced Products
- Scalar Triple Product a·(b×c)
- Volume of Parallelepiped
- Vector Triple Product Preview
CELL 41-43
Direction
- Direction Cosines (l, m, n)
- l² + m² + n² = 1
- Direction Ratios
- Angles with Coordinate Axes
CELL 44-52
3D Geometry Bridge
- Section Formula with Vectors
- Vector Equation of a Line
- Parametric & Cartesian Forms
- Equation of a Plane (Preview)
- Angle & Skew Lines (Preview)
CELL 53-57
Physics Applications
m
- Position, Velocity, Acceleration
- Force (F = ma)
- Momentum (p = mv)
- Electric Field
CELL 58-62
Higher Mathematics
- Gradient & Vector Fields
- Divergence & Curl (Frontier)
- Basis Vectors
- High-Dimensional Data Preview
CELL 63-67
The Vector Synthesis
- Geometry → Vector
- Vector → Algebra
- Vector → Physics
- The Vector World Mastered
Exit Portal
We now understand how vectors point, move, and act through space. The coordinate system tells us where the body is. The vector tells us where it is going and how strongly something acts upon it. We are now ready to systematically build the full geometry of lines, planes, angles, distances, and shortest paths in 3D space.