One Vector Its Components And Its Basis
This animation shows a single physical vector arrow decomposed into components along a coordinate basis, illustrating that A = A1e1 + A2e2 + A3e3. It demonstrates how the reciprocal basis acts as a measuring tool to extract each component (A1 = A·e1), and introduces the metric gji = ej·ei as a way to describe how basis directions overlap. Useful for students beginning vector calculus or tensor analysis who need an intuitive, visual grounding before facing index notation.
16:9 · Preview before teaching · automatic layout checks do not establish subject accuracy
The prompt that made it
Create a visual titled: “2(a) — One Vector, Its Components, and Its Basis” Follow the MASTER INSTRUCTION. Scene 1 Show a physical arrow A. Say: “This arrow is the real physical object.” Scene 2 Put a coordinate grid around it. Break A into three component arrows. Show: A = A1e1 + A2e2 + A3e3 Explain: “The components are simply the amounts of each basis direction needed to rebuild A.” Scene 3 Show how to measure A1. Use the reciprocal basis as a measuring tool: A1 = A · e1 Repeat visually for A2 and A3. Scene 4 Explain the metric: gji = ej · ei as: “The metric tells us how the basis directions overlap.” Do not introduce complicated index manipulation. End: A vector is an arrow. Components are the numbers needed to rebuild the arrow.
Make the next one in this series
- Show two different bases describing the same vector A with different components
- Animate how the reciprocal basis vectors are constructed geometrically from the original basis
- Demonstrate how the metric tensor changes when the basis is non-orthogonal versus orthogonal
