# Is this a scalar product of three vectors?

In Arnold's Mathematical Methods of Classical Mechanics, page 179, he says:

“In an oriented euclidean three-space every vector $$\mathbf{A}$$ corresponds to a 1-form $$\omega_{\mathbf{A}}^{1}$$ and a 2-form $$\omega_{\mathbf{A}}^{2}$$ defined by the conditions$$\omega_{\mathbf{A}}^{1}\left(\mathbf{\xi}\right)=\left(\mathbf{A},\xi\right)$$ $$\omega_{\mathbf{A}}^{2}\left(\mathbf{\xi,\eta}\right)=\left(\mathbf{A},\xi,\eta\right)$$ $$\xi,\eta\in\mathbb{R}^{3}.”$$

I take $$\left(\mathbf{A},\xi\right)$$ to be a scalar product. So he's saying the 1-form $$\omega_{\mathbf{A}}^{1}$$ acts on the vector $$\xi$$ to give the same number as the scalar product $$\left(\mathbf{A},\xi\right)$$. But what does $$\left(\mathbf{A},\xi,\eta\right)$$ mean? Is it some sort of scalar product of three vectors (multiplying out the respective components)? Thanks.

EDIT

I understand that the exterior product of two 1-forms is given by$$\left(\omega_{1}\wedge\omega_{2}\right)\left(\xi,\eta\right)=\left|\begin{array}{cc} \omega_{1}\left(\xi\right) & \omega_{1}\left(\eta\right)\\ \omega_{2}\left(\xi\right) & \omega_{2}\left(\eta\right) \end{array}\right|.$$ But I can't see why this is equal to the determinant of $$\left(\mathbf{A},\xi,\eta\right).$$

• For three 3-vectors, there is the triple product $a\cdot(b \times c)$. – achille hui Jul 15 '19 at 15:03

In the standard $$(x,y,z)$$ coordinates, write $$\mathbf A = (a,b,c)$$ and let $$\omega^2_{\mathbf A} = a\,dy\wedge dz + b\,dz\wedge dx + c\,dx\wedge dy$$. Now check, expanding in cofactors along the first column, that $$\omega^2_{\mathbf A}(\xi,\eta) = \det(\mathbf A,\xi,\eta).$$
• It works! I get $\omega_{\mathbf{A}}^{2}(\xi,\eta)=a\left(\xi^{2}\eta^{3}-\xi^{3}\eta^{2}\right)+b\left(\xi^{3}\eta^{1}-\xi^{1}\eta^{3}\right)+c\left(\xi^{1}\eta^{2}-\xi^{2}\eta^{1}\right),$ which is equal to $\det(\mathbf{A},\xi,\eta)$. Thanks. – Peter4075 Jul 16 '19 at 7:20
Arnold's product of $$3$$ vectors is simply the determinant of the respective $$3\times3$$-matrix.