# Calculating the cross product of a cross product

so I really can't see what I am doing wrong. I want to use this formula:

$$a\times (b\times c) = b(a\cdot c) - c(a\cdot b)$$

Calculate the rotation of $$v(x,y,z)=(x,y,z)^T \times \omega$$ with $$\omega \in \mathbb R^3$$

Solution:

$$a\cdot c=\nabla\cdot \omega=0$$

and

$$a\cdot b = \nabla \cdot (x,y,z)^T= \partial_x x + \partial_y y + \partial_z z = 3$$

so we get $$-\omega 3=-3\omega$$

The actual solution (which I do get by direct calculation) is: $$-2\omega$$

If you want to generalise formulae with vectors having commuting components, make sure to write the usual formula so as to preserve the order in products. For example, here the $$i$$th component of the RHS is $$\sum_j a_j (b_i c_j-b_j c_i)$$, once we impose the $$abc$$ order of the LHS. Now you know the generalisation off the top of your head. For $$a_i=\partial_i$$, the result's $$i$$th component is $$\sum_j \partial_j (b_i c_j-b_j c_i)=b_i\nabla\cdot c+c\cdot\nabla b_i-(\nabla\cdot b) c-b\cdot\nabla c_i.$$The vector, in other words, is $$\vec{b}(\nabla\cdot\vec{c})-(\nabla\cdot\vec{b})\vec{c}+(\vec{c}\cdot\nabla)\vec{b}-(\vec{b}\cdot\nabla)\vec{c}.$$(I've swapped the middle terms to mirror @md2perpe's quoted result, but the letters $$b,\,c$$ still need to be changed to $$A,\,B$$).
On Wikipedia you can see that the formula for the curl of a cross product is given by $$\nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A}\ (\nabla \cdot \mathbf{B}) - \mathbf{B}\ (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} .$$
Applying this on your case gives \begin{align} \nabla \times (\mathbf{x} \times \mathbf{\omega}) &= \mathbf{x}\ (\nabla \cdot \mathbf{\omega}) - \mathbf{\omega}\ (\nabla \cdot \mathbf{x}) + (\mathbf{\omega} \cdot \nabla) \mathbf{x} - (\mathbf{x} \cdot \nabla) \mathbf{\omega} \\ &= \mathbf{x} \ 0 - \mathbf{\omega}\ 3 + \mathbf{\omega} - \mathbf{0} = -2 \mathbf{\omega} . \end{align}