Dot product of a vector and del operator How did this expansion come about.

What is the physical significance of this expansion.
And what is the significance of 
$$\vec{A} . \nabla {\vec{A}}$$
And what does this mathematically represent.
 A: $\newcommand{\col}[3]{\left(\begin{matrix} #1 \\ #2 \\ #3 \end{matrix}\right)}$
$\newcommand{\vecc}[1]{\col{#1_x}{#1_y}{#1_z}}$
$$
\vec A \times (\nabla \times \vec A)
= \vecc A \times \left( \vecc \partial \times \vecc A \right)
= \vecc A \times \col{\partial_y A_z - \partial_z A_y}{\partial_z A_x - \partial_x A_z}{\partial_x A_y - \partial_y A_x} \\
= \col
{A_y(\partial_x A_y - \partial_y A_x) - A_z(\partial_z A_x - \partial_x A_z)}
{A_z(\partial_y A_z - \partial_z A_y) - A_x(\partial_x A_y - \partial_y A_x)}
{A_x(\partial_z A_x - \partial_x A_z) - A_y(\partial_y A_z - \partial_z A_y)}
= \col
{(A_y \partial_x A_y + A_z \partial_x A_z) - (A_y \partial_y + A_z \partial_z) A_x}
{(A_z \partial_y A_z + A_x \partial_y A_x) - (A_z \partial_z + A_x \partial_x) A_y}
{(A_x \partial_z A_x + A_y \partial_z A_y) - (A_x \partial_x + A_y \partial_y) A_z} \\
= \col
{(A_x \partial_x A_x + A_y \partial_x A_y + A_z \partial_x A_z) - (A_x \partial_x + A_y \partial_y + A_z \partial_z) A_x}
{(A_x \partial_y A_x + A_y \partial_y A_y + A_z \partial_y A_z) - (A_x \partial_x + A_y \partial_y + A_z \partial_z) A_y}
{(A_x \partial_z A_x + A_y \partial_z A_y + A_z \partial_z A_z) - (A_x \partial_x + A_y \partial_y + A_z \partial_z) A_z}
= \col
{\vec A \cdot \partial_x \vec A - (\vec A \cdot \nabla) A_x}
{\vec A \cdot \partial_y \vec A - (\vec A \cdot \nabla) A_y}
{\vec A \cdot \partial_z \vec A - (\vec A \cdot \nabla) A_z}
= \col
{\frac12 \partial_x (\vec A \cdot \vec A)}
{\frac12 \partial_y (\vec A \cdot \vec A)}
{\frac12 \partial_z (\vec A \cdot \vec A)}
- \col
{(\vec A \cdot \nabla) A_x}
{(\vec A \cdot \nabla) A_y}
{(\vec A \cdot \nabla) A_z}
\\
= \frac12 \nabla (\vec A \cdot \vec A) - (\vec A \cdot \nabla) \vec A
$$
Thus,
$$(\vec A \cdot \nabla) \vec A = \frac12 \nabla (\vec A \cdot \vec A) - \vec A \times (\nabla \times \vec A)$$
