Wedge Product - Exterior Product (𝑣∧𝑢)
- is a multilinear algebra operation that produces a bivector from two vectors
- the wedge product ∧ of two vectors 𝑣,𝑢 is represented as 𝑣∧𝑢
- is used to construct the geometric product
- is similar to the outer product
- is similar to the cross product
Wedge Product - 3D Definition
Given two vectors 𝑣,𝑢 in ℝ3:
- 𝑣 = 𝑣1e1 + 𝑣2e2+ 𝑣3e3
- 𝑢 = 𝑢1e1 + 𝑢2e2+ 𝑢3e3
the wedge product 𝑣∧𝑢 is defined as:
- 𝑣∧𝑢 = (𝑣2𝑢3- 𝑣3𝑢2)e2e3+ (𝑣3𝑢1- 𝑣1𝑢3)e3e1+ (𝑣1𝑢2- 𝑣2𝑢1)e1e2
the wedge product 𝑣∧𝑢 can be represented as a bivector (which is syntactically similar to a vector):
- where each element in this vector is a scalar that represents the magnitude in the associated direction
- (e.g. scalar 𝑣2𝑢3- 𝑣3𝑢2in direction e2e3 which is part of a basis bivector)
Wedge Product - General Definition
TODO