Same Arrow, Completely Different Numbers
A single fixed arrow is measured twice: once against an upright coordinate grid, then again after swapping in a tilted second grid. The component numbers change dramatically, yet the arrow itself never moves. The animation contrasts the physical vector with its coordinate description, introducing notation A = Ai ei versus A = Āi gi and the idea of a second metric as a new measuring table. Useful for students learning tensor notation, change of basis, and why components are observer-dependent while vectors are not.
16:9 · Preview before teaching · automatic layout checks do not establish subject accuracy
The prompt that made it
Create a visual titled: “2(b) — Same Arrow, Completely Different Numbers” Follow the MASTER INSTRUCTION. Scene 1 Draw one large physical arrow A. Freeze it visually. Add coordinate system #1. Find its components. Scene 2 Keep A EXACTLY where it is. Remove coordinate system #1. Replace it with a tilted coordinate system #2. Find the new components. Make the numerical values visibly different. Scene 3 Ask: “Did the vector change?” Big answer: NO. Only our description changed. Scene 4 Show: A = Ai ei and: A = Āi gi Explain that the bar over the components means: “numbers belonging to the second basis.” Scene 5 Introduce the second metric as the new system's measuring table. End: Different coordinate systems can give different numbers for the same physical arrow.
Make the next one in this series
- Show how the same vector's length stays invariant even though components change between bases
- Animate how basis vectors ei and gi themselves relate through a transformation matrix
- Demonstrate covariant versus contravariant components using the two tilted coordinate systems
