Need help understanding vector identity

Question

Been trying to figure this out for hours now and finally came here for help. According to Wikepedia: $\mathbf{A}\times(\nabla\times\mathbf{B}) = (\nabla\mathbf{B})\cdot\mathbf{A} - \mathbf{A}\cdot(\nabla\mathbf{B})$. Can someone convince me how $(\nabla\mathbf{B})\cdot\mathbf{A} \neq \mathbf{A}\cdot(\nabla\mathbf{B})$? Because the dot product is commutative, so shouldn't it just equal $0$? Thanks