e1e2

e1e2 is a notation that may refer to the product of the standard basis vectors e1 and e2 in a two-dimensional space. In many mathematical contexts, e1 and e2 denote an orthonormal basis of R2 or of a 2-dimensional subspace. The juxtaposition e1e2 often appears as the geometric product in geometric algebra or as the exterior (wedge) product e1 ∧ e2. The two interpretations are related but distinct: the exterior product e1 ∧ e2 is a bivector representing the oriented area spanned by e1 and e2, whereas the geometric product e1e2 encodes both area and orientation and can be used to compose rotations and other transformations.

In 2D Euclidean geometric algebra, e1e2 is a unit bivector with the property e1e2 = - e2e1 and (e1e2)^2 = -1. This makes it analogous to the imaginary unit i in complex numbers, and rotations in the plane can be expressed as exponentials of this bivector, for example e^{θ e1e2}.

Generalization: In higher dimensions, e1e2 is one of many simple bivectors e_i e_j (i < j) that form a basis for the space of 2-blades. The object e1e2 appears in differential forms, computer graphics, and physics to represent the plane spanned by the axes corresponding to e1 and e2, and as a building block for more complex transformations.

Ambiguity: Without additional context or a wedge symbol, e1e2 can be interpreted in multiple ways. The intended meaning is usually clear from surrounding notation, such as whether a wedge or a geometric product is being implied.