The wedge product turns area and volume into algebra with a sign
Take two arrows on a flat page and they outline a parallelogram. Hermann Grassmann's wedge product multiplies the two arrows and gets that area directly, with a plus or minus sign recording which way round the arrows go. Swap them and the sign flips. Multiply an arrow by itself and you get zero.
The exterior algebra, also called Grassmann algebra, was built as a tool for studying areas, volumes and their higher-dimensional versions. Wedge two vectors together and you get a 2-blade, whose size is the area of their parallelogram. Wedge k vectors and you get a k-blade, measuring the volume of the slanted box they span. The name comes from the wedge-shaped symbol used for the product.
A worked example shows why it behaves this way. Write two flat vectors in terms of perpendicular unit arrows and expand their wedge product using ordinary distribution. Terms pairing an arrow with itself vanish, and swapping the order of two arrows introduces a minus sign. What survives is ad minus bc times a single unit piece, exactly the determinant familiar from school formulas for parallelogram area.
That is no coincidence. Try listing the rules any sensible notion of signed area must obey: stretching a side stretches the area by the same factor; a collapsed parallelogram, both sides pointing the same way, has zero area; swapping the sides reverses orientation; and sliding one side along the direction of the other changes neither base nor height. The wedge product satisfies all of these. The one thing it adds is freedom from any chosen unit square, giving a description of area that does not depend on coordinates.
Beyond single blades, the algebra contains sums of them, called multivectors, organised by degree. It works over any field of numbers and even more general settings, and the calculus of differential forms is itself an exterior algebra. In three dimensions it is closely related to the familiar cross product and triple product.
Source: Exterior algebra
