🧄Proving Contravariant Vector Components Using the Dual Basis

A vector's contravariant components are found by projecting the vector onto the dual basis vectors. This process, governed by the elegant mathematical relationship $v^a=E^a \cdot v$, reveals a fundamental symmetry in tensor analysis: just as covariant components are projections onto the standard basis, contravariant components are projections onto the dual basis. The visual demo confirms this abstract relationship, showing that the formula is a direct geometric representation of vector decomposition.

🎬how an arbitrary vector can be decomposed into its components by taking the dot product with the dual basis vectors

The visualization demonstrates that the formula isn't just an abstract equation. It has a clear, geometric meaning: a vector's contravariant component is simply the length of its projection onto the corresponding dual basis vector. The demo makes this concept clear by visually showing the projection lines, proving that the math is directly reflected in the geometry.

🖊️Mathematical Proof