Wedge & Star
The wedge sweeps an oriented area. The star keeps everything you left out.
Grassmann's product builds areas and volumes directly out of vectors; Hodge's star hands each one its orthogonal complement. Drag the arrow tips. Swap the order and watch the circulation flip. Then add a fourth axis and watch the dual of a plane stop being an arrow.
⋆(u ∧ v) = u × v — the cross product
u ∧ v is not a number. It is an area that remembers its circulation.
u ∧ u = 0: sweeping along yourself encloses nothing.
In three dimensions, ⋆ turns a plane into its normal. That is the entire cross product.
In four, the dual of a plane is another plane. The axis was a coincidence.
The star, dimension by dimension
Euclidean metric, standard orientation. Each blade maps to the blade it leaves out, with a sign that keeps e₁∧…∧eₙ positive. The middle column of ℝ³ is the entire vector-calculus toolkit; the middle column of ℝ⁴ is why that toolkit stops there.
ℝ²
the star is a quarter turn
⋆1=e₁₂
⋆e₁=e₂
⋆e₂=−e₁
⋆e₁₂=1
ℝ³
planes trade places with axes
⋆e₁=e₂₃
⋆e₂=−e₁₃
⋆e₁₂=e₃
⋆e₁₂₃=1
ℝ⁴
planes trade places with planes
⋆e₁=e₂₃₄
⋆e₁₂=e₃₄
⋆e₁₃=−e₂₄
⋆e₁₂₃=e₄