$\displaystyle \nabla X (\vec A X \vec B) \;=\; (\vec B \cdot \nabla)\vec A - \vec B(\nabla \cdot \vec A) -(\vec A \cdot \nabla)\vec B + \vec A ( \nabla \cdot \vec B) $.
What is $\displaystyle \vec A \cdot \nabla $?
It's the vector $\displaystyle \vec{A}$ dotted with the gradient operator. The result of this dot product is a scalar operator. So, for example, the expression
$\displaystyle \displaystyle(\vec{A}\cdot\nabla)\vec{B}=\left(\su m_{j=1}^{3}A_{j}\,\dfrac{\partial}{\partial x_{j}}\right)\vec{B}=\left\langle\sum_{j=1}^{3}A_{ j}\,\dfrac{\partial B_{1}}{\partial x_{j}},\sum_{j=1}^{3}A_{j}\,\dfrac{\partial B_{2}}{\partial x_{j}},\sum_{j=1}^{3}A_{j}\,\dfrac{\partial B_{3}}{\partial x_{j}}\right\rangle.$
Does that make sense?