Curl of product

Question

$$\nabla \times (A\times B) = A(\nabla \cdot B)-B(\nabla\cdot A) + (B\cdot \nabla)A - (A\cdot \nabla)B.$$

What's the difference between $B(\nabla\cdot A)$ and $(A\cdot \nabla)B$?

Say $A= (a_1,a_2)$ and $B=(b_1,b_2)$.

Then $B(\nabla\cdot A)$ is

$\begin{pmatrix} b_1 (a_1)_x +b_1 (a_2)_y \\ b_2 (a_1)_x +b_2 (a_2)_y \end{pmatrix}$

What about $(A\cdot \nabla)B$? If it means $A\cdot (\nabla B)$, then I will get a scalar, so I can't see how it can be added to $B(\nabla\cdot A)$, which is a vector.

If it means $$(A\cdot \langle \partial_x,\partial_y\rangle)B = \langle(a_1)_x + (a_2)_y\rangle B,$$ then I don't see how it differs from $B(\nabla\cdot A).$

Answer 1

Note that $$ A \cdot \nabla = a_1 \partial_x + a_2 \partial_y \ne (a_1)_x + (a_2)_y = \nabla \cdot A $$ So $$ (A \cdot \nabla)B = \binom{a_1 (b_1)_x + a_2(b_1)_y}{a_1(b_2)_x + a_2(b_2)_y} $$

Answer 2

$(A\cdot\nabla)B$ is the vector $$\left(A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y}+A_z\frac{\partial}{\partial z}\right)B_x \hat{x}+\left(A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y}+A_z\frac{\partial}{\partial z}\right)B_y \hat{y}+\left(A_x \frac{\partial}{\partial x}+A_y \frac{\partial}{\partial y}+A_z\frac{\partial}{\partial z}\right)B_z \hat{z}$$ while $B(\nabla\cdot A)$ is the vector $$\left(\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\right)\left(B_x \hat{x} + B_y\hat{y}+B_z\hat{z}\right)$$