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The Wedge product is the multiplication operation in exterior algebra. The wedge product is always antisymmetric, associative, and anti-commutative. The result of the wedge product is known as a bivector; in \R^3 (that is, three dimensions) it is a 2-form. For two vectors u and v in ℝ3, the wedge product is defined as

\mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} =

\begin{bmatrix}
0 & u_1 v_2 - u_2 v_1 & u_1 v_3 - u_3 v_1  \\
u_2 v_1 - u_1 v_2 & 0 & u_2 v_3 - u_3 v_2 \\
u_3 v_1 - u_1 v_3 & u_3 v_2 - u_2 v_3 & 0
\end{bmatrix}

where ⊗ denotes the outer product. Note that the bivector has only three indepedent elements; as such, it can be associated with another vector in ℝ3. If the associated vector is defined as

\mathbf{n}=

\begin{bmatrix}
(\mathbf{u} \wedge \mathbf{v} )_{23}  \\
-(\mathbf{u} \wedge \mathbf{v} )_{13} \\
(\mathbf{u} \wedge \mathbf{v} )_{32}
\end{bmatrix}

it is the same as the cross product of u and v. In this sense, the cross product is a special case of the exterior product.

Bivectors

See also: Rotation_matrix#Rotation_matrix_from_axis_and_angle and Cross_product#Conversion_to_matrix_multiplication

Bivectors are commonly used to represent rotations.


See also

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