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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 ℝ3 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.