Vectors
VECTOR LAB
2D DYNAMICS VISUALIZER
Origin Centered at (200, 200)
Vector Flow
Magnitude & Direction. In the world of physics, numbers aren't enough. We need to know Where the energy is headed.
- ๐ Components: Broken down into x and y coordinates.
- ๐๏ธ Velocity: Speed combined with a specific heading.
- ๐น Displacement: The straight-line path from start to finish.
Metric Scale
Absolute Size. The magnitude strips away the direction to reveal the Pure Quantity of the vector.
- ๐ Length: Calculated as the square root of the sum of squares.
- ๐ Scalar: The result is always a single positive number.
- ๐๏ธ Speed: Magnitude turns Velocity vectors into Speed.
Projection Core
Scalar Alignment. The dot product A ยท B determines how much of one vector exists in the direction of another.
- ๐ Orthogonality: If the result is 0, the vectors are perfectly perpendicular.
- ๐ง Similarity: Used in AI to find related concepts in vector space.
- ๐๏ธ Work: Calculates energy transfer when force and motion align.
Orthogonal Hub
Dimensional Shift. The cross product a x b generates a vector that exists in a New Dimension perpendicular to the input plane.
- โ Direction: Determined strictly by the Right-Hand Rule.
- ๐ง Torque: The math behind rotational force and leverage.
- ๐ฎ Surface Normals: Crucial for 3D rendering and light physics.
Unit Core
Standardized Path. The unit vector รข = a / |a| represents the Pure Direction of motion, with a magnitude of exactly 1.0.
- โญ Unit Circle: All unit vectors lie on the edge of a circle with radius 1.
- ๐งญ Direction Only: Ignores strength to focus on heading.
- ๐ ๏ธ Normalization: The core process of dividing a vector by its length.
Angular Insight
Relative Orientation. Determining the angle ฮธ reveals how two forces or paths interact within a Vector Field.
- ๐ Cosine Rule: Uses the Dot Product to find the gap between vectors.
- ๐ข Orthogonality: Instantly detects if vectors are at a perfect 90ยฐ.
- ๐ Flight Paths: Crucial for calculating orbital entry and steering.
Shadow Metric
Component Logic. The scalar projection (a · b) / |b| extracts the Effective Magnitude of one vector along another path.
- ๐ Directional Focus: Ignores the perpendicular part of vector A.
- ๐ Length Only: Results in a scalar value, not a vector.
- ๐ Effective Force: Used to find the component of gravity acting down a slope.
Vectors (v)
v = xi + yj
Magnitude = โ(xยฒ + yยฒ)
Direction = ฮธ = tanโปยน(y/x)
Examples = Force, Velocity
PHET VECTOR ADDITION
Drag and drop arrows to see how "Head-to-Tail" vector addition works visually.
PLAY SIMPHYSICS CLASSROOM
Master the difference between distance/displacement and speed/velocity.
READ BASICSGEO-GEBRA VECTORS
Create and manipulate 3D vectors to understand magnitude and components.
OPEN TOOL
