exterior product

not to be confused with outer product. Note that the exterior product is sometimes called outer product in literatures.

The exterior product, also called wedge product, of vectors and constructs a bivector representing the parallelogram formed by the vectors    and  , in the plane they span together. ¹ We can also perform the wedge product of three vectors, or a vector and a bivector, to form a trivector . In general, the wedge products of vectors are called a -blade (simple -vector).

Exterior product is closely related to the cross product:

Properties

• (anti-commutativite)
• Distributive over addition:

Magnitude

The magnitude of exterior product is the same as the magnitude cross product.

Theorem

Let be an orthonormal basis for a plane. Orient the plane with that of . Let and be vectors in the plane. Let be the oriented angle from to , . Then

In 3D

So far we are talking about the coordinate-free definition of exterior product, and here is a definition with three coordinate vectors:

Let and . Then

and

Related

• inner product
• outer product
• cross product
• geometric product
• Cartesian product
• Hadamard product - element-wise product

Footnotes

1. Let’s remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra) - Marc ten Bosch ↩