Editing Exterior algebra (section)

===Cross and triple products===
[[File:2-vector decomposition.png|thumb|right|Basis Decomposition of a 2-vector]]
For vectors in '''R'''<sup>3</sup>, the exterior algebra is closely related to the [[cross product]] and [[triple product]]. Using the standard basis {{nowrap|{{mset|'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>}}}}, the exterior product of a pair of vectors
: <math> \mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 </math>
and
: <math> \mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + v_3 \mathbf{e}_3 </math>
is
: <math> \mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) (\mathbf{e}_1 \wedge \mathbf{e}_2) </math>
: <math> \phantom{\mathbf{u} \wedge \mathbf{v}} + (u_3 v_1 - u_1 v_3) (\mathbf{e}_3 \wedge \mathbf{e}_1) </math>
: <math> \phantom{\mathbf{u} \wedge \mathbf{v}} + (u_2 v_3 - u_3 v_2) (\mathbf{e}_2 \wedge \mathbf{e}_3) </math>
where {{mset|'''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub>, '''e'''<sub>3</sub> ∧ '''e'''<sub>1</sub>, '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub>}} is the basis for the three-dimensional space ⋀<sup>2</sup>('''R'''<sup>3</sup>). The coefficients above are the same as those in the usual definition of the [[cross product]] of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a [[bivector]].

Bringing in a third vector
: <math> \mathbf{w} = w_1 \mathbf{e}_1 + w_2 \mathbf{e}_2 + w_3 \mathbf{e}_3, </math>
the exterior product of three vectors is
: <math> \mathbf{u} \wedge \mathbf{v} \wedge \mathbf{w} = (u_1 v_2 w_3 + u_2 v_3 w_1 + u_3 v_1 w_2 - u_1 v_3 w_2 - u_2 v_1 w_3 - u_3 v_2 w_1) (\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3) </math>
where '''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub> is the basis vector for the one-dimensional space ⋀<sup>3</sup>('''R'''<sup>3</sup>). The scalar coefficient is the [[triple product]] of the three vectors.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product {{nowrap|'''u''' × '''v'''}} can be interpreted as a vector which is perpendicular to both '''u''' and '''v''' and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the [[minor (mathematics)|minors]] of the matrix with columns '''u''' and '''v'''. The triple product of '''u''', '''v''', and '''w''' is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns '''u''', '''v''', and '''w'''. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively oriented [[orthonormal basis]], the exterior product generalizes these notions to higher dimensions.