Project 16: The Vector Geometry Toolkit

Build a vector toolkit that computes magnitude, angle, projection, and decomposition with explicit geometric interpretation.

Quick Reference

Attribute Value
Difficulty Intermediate-Advanced (Level 2-3)
Time Estimate 10-18 hours
Main Programming Language Python
Alternative Programming Languages C++, Julia, JavaScript
Key Topics Norms, dot product, projections, orthogonality
Input Mode CLI vector strings
Output Mode Numeric diagnostics + optional vector plots

1) Learning Objectives

  1. Implement vector operations with dimension-safe checks.
  2. Interpret dot product as angle/projection information.
  3. Decompose vectors into parallel and orthogonal components.
  4. Verify geometric invariants numerically.
  5. Communicate numeric results as geometric stories.

2) All Theory Needed (Per-Concept Breakdown)

Concept A: Vectors as Structured Quantities

A vector is not just a list of numbers; it is a quantity with coordinate system, magnitude, and direction. Operations only make sense for compatible dimensions.

Concept B: Dot Product and Geometry

Dot product measures directional alignment. It powers angle computation and projection formulas, connecting algebra to geometry.

Concept C: Decomposition and Orthogonality

Any vector can be split into a component parallel to a reference direction and a component orthogonal to it. This decomposition is foundational in physics, graphics, and optimization.

3) Project Specification

3.1 What You Will Build

A CLI toolkit supporting:

  1. Norm and unit-vector computation.
  2. Dot-product and angle calculations.
  3. Projection and orthogonal decomposition.
  4. Invariant checks (recomposition error, orthogonality residual).

3.2 Functional Requirements

  1. Parse 2D and 3D vectors.
  2. Reject incompatible vector dimensions.
  3. Return angle in degrees and radians.
  4. Report projection and rejection vectors.
  5. Emit validation metrics in a deterministic format.

3.3 Non-Functional Requirements

  • Stable numeric precision.
  • Clear handling of zero vectors.
  • Deterministic field ordering.

3.4 Real World Outcome

$ python vector_toolkit.py --v "3,4" --u "1,0" --mode all
[norm] ||v|| = 5.0000
[dot] v.u = 3.0000
[angle_deg] 53.1301
[projection_on_u] (3.0000, 0.0000)
[orth_component] (0.0000, 4.0000)
[check] projection + orth = v (error=0.0000)

$ python vector_toolkit.py --v "1,2,3" --u "2,-1,0" --mode angle
[result] angle_deg = 72.4516

4) Solution Architecture

4.1 High-Level Design

Input Parser -> Dimension Validator -> Vector Ops Engine -> Invariant Checker -> Reporter

4.2 Key Components

Component Responsibility
Vector Parser Convert CLI text into numeric vectors
Ops Engine Compute norm/dot/angle/projection/decomposition
Invariant Checker Validate recomposition and orthogonality
Reporter Present results in reproducible format

5) Implementation Guide

Phase 1: Core Operations

  • Build norm and dot operators.
  • Add dimension and zero-vector guards.
  • Add deterministic formatting for outputs.

Phase 2: Angle and Projection

  • Implement angle calculation with cosine clamping.
  • Add projection/rejection decomposition.
  • Add explanatory labels for geometric interpretation.

Phase 3: Validation and Visualization

  • Compute recomposition error.
  • Add optional vector plot output.
  • Create test fixtures with known answers.

6) Validation Checklist

  • Dot and norm values match hand calculations.
  • Angle computation remains stable near parallel vectors.
  • Projection + orthogonal component reconstruct original vector.
  • Dimension mismatch and zero-vector cases are handled gracefully.

7) Extension Ideas

  1. Add cross-product support.
  2. Add basis change and coordinate-transform modes.
  3. Add 3D interactive visualization export.

8) Books and References

  • Math for Programming - vectors and linear algebra foundations.
  • Introduction to Computation and Programming Using Python - numerical checks.
  • Matplotlib vector plotting documentation.