The problem is concerning to develop the correct algebra applying the calculus properties. The question is in the figure embedded.
We have the following Helmholtz (vector-scalar) equation given by
\begin{align} \frac{\omega^2}{c^2}\phi + \left( \nabla - i \vec{A}\right)^2 \phi=0 \end{align}
where $\vec{A}$ is a vector and $\phi$ is scalar function.
Of course, the second term is square also this a vector operator.
The question is how to develop the right algebra for the second term ? \begin{align*} \left( \nabla - i \vec{A}\right) \left( \nabla - i \vec{A}\right) \phi \end{align*} or \begin{align*} \left( \nabla - i \vec{A}\right) \cdot \left( \nabla - i \vec{A}\right) \phi \end{align*}
We can make at first sight the analogy $(a+b)^2 = (a+b)(a+b)$, valid for real numbers or the vectorial case version, but here scalar and vector (funtions) are mixed.
