Consider now the wedge product of the 1-forms \begin{equation} v = v_x\,dx + v_y\,dy + v_z\,dz \end{equation} and \begin{equation} w = w_x\,dy\wedge dz + w_y\,dz\wedge dx + w_z\,dx\wedge dy \end{equation} This is a 3-form, and there is only one independent 3-form, so each term is a multiple of every other. Adding them up, we obtain \begin{equation} v\wedge w = (v_xw_x+v_yw_y+v_zw_z) \,dx\wedge dy\wedge dz \end{equation} which looks very much like the dot product between $\vv$ and $\ww$! But the dot product is a scalar, so we really want just the coefficient of our basis 3-form $dx\wedge dy\wedge dz$.

As with the cross product, a byproduct of our analysis of the dot product is to establish an implicit map, this time between the 0-forms and 3-forms on $\RR^3$, as briefly discussed in § Relationships between Differential Forms; such identifications will be discussed further in Chapter 3.