wedge (exterior) product

The wedge product v ^ w combines two vectors into an antisymmetric object that measures the oriented area of the parallelogram they span. Its one law is antisymmetry: v ^ w = -(w ^ v). Setting w = v forces v ^ v = 0, which says a degenerate parallelogram (two equal sides) has zero area — a fact you already believe geometrically.

More generally a wedge of k vectors v1 ^ v2 ^ ... ^ vk is the oriented k-volume of the parallelepiped they span. Swapping any two factors flips the sign, and repeating any factor sends the whole thing to zero. So linear dependence is detectable by wedging: vectors are dependent exactly when their wedge vanishes.

In coordinates the wedge collects the antisymmetric combinations. For instance in R^3, (a ^ b) read in the basis e_i ^ e_j gives the components of the cross product a x b. And v1 ^ ... ^ vn for an n-by-n system equals det of the matrix times the volume form e1 ^ ... ^ en — the wedge is where the determinant is born.

Be careful: the wedge is graded-commutative, not commutative. A degree-p form alpha and a degree-q form beta satisfy alpha ^ beta = (-1)^(p*q) * (beta ^ alpha). Two 1-forms anticommute (the (-1)^1 case), but a 2-form commutes past a 2-form. Keeping track of these signs is the everyday discipline of working with differential forms.