wedge product
The wedge product is multiplication that remembers orientation and refuses repetition. When you wedge two vectors you build the oriented parallelogram they span; swapping the order flips the orientation and the sign, and wedging a vector with itself gives zero because a degenerate parallelogram has no area. It is the product that powers the exterior algebra and signed-volume calculations.
Formally ∧ is the multiplication of the exterior algebra Λ(V), inherited from concatenation in the tensor algebra after imposing v ∧ v = 0. The defining identities are bilinearity in each slot, antisymmetry v ∧ w = −w ∧ v, and more generally the graded-commutative rule: if α has degree p and β has degree q then α ∧ β = (−1)^{pq} β ∧ α. The wedge of a p-form and a q-form lands in degree p + q.
A wedge v_1 ∧ ... ∧ v_k vanishes exactly when v_1, ..., v_k are linearly dependent, so nonzero decomposable wedges correspond to k-dimensional subspaces with an orientation and a scale — this is the geometric content exploited by Plücker coordinates and Grassmannians. The wedge is also the cornerstone of the calculus of differential forms and the operation underlying the determinant.
In R^2, (a e_1 + b e_2) ∧ (c e_1 + d e_2) = (ad − bc) e_1 ∧ e_2, since e_1 ∧ e_1 = e_2 ∧ e_2 = 0 and e_2 ∧ e_1 = −e_1 ∧ e_2. The coefficient ad − bc is exactly the 2 × 2 determinant — the signed area.
The wedge of two plane vectors reads off the determinant ad − bc.