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.
- 📐 Components: Broken down into x and y coordinates.
- 🏎️ Velocity: Speed combined with a specific heading.
- 🏹 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.
- 📏 Length: Calculated as the square root of the sum of squares.
- 💎 Scalar: The result is always a single positive number.
- 🏎️ Speed: Magnitude turns Velocity vectors into Speed.
Projection Core
Scalar Alignment. The dot product A · B determines how much of one vector exists in the direction of another.
- 📐 Orthogonality: If the result is 0, the vectors are perfectly perpendicular.
- 🧠 Similarity: Used in AI to find related concepts in vector space.
- 🏗️ 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.
- ✋ Direction: Determined strictly by the Right-Hand Rule.
- 🔧 Torque: The math behind rotational force and leverage.
- 🎮 Surface Normals: Crucial for 3D rendering and light physics.
Unit Core
Standardized Path. The unit vector â = a / |a| represents the Pure Direction of motion, with a magnitude of exactly 1.0.
- ⭕ Unit Circle: All unit vectors lie on the edge of a circle with radius 1.
- 🧭 Direction Only: Ignores strength to focus on heading.
- 🛠️ Normalization: The core process of dividing a vector by its length.
Angular Insight
Relative Orientation. Determining the angle θ reveals how two forces or paths interact within a Vector Field.
- 📐 Cosine Rule: Uses the Dot Product to find the gap between vectors.
- 🟢 Orthogonality: Instantly detects if vectors are at a perfect 90°.
- 🚀 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.
- 📐 Directional Focus: Ignores the perpendicular part of vector A.
- 📏 Length Only: Results in a scalar value, not a vector.
- 🚀 Effective Force: Used to find the component of gravity acting down a slope.
Vectors (v)
v = xi + yj
Magnitude = √(x² + y²)
Direction = θ = tan⁻¹(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
