math-mode: \vec and vector vector norm: Vector arrownot covered

Question

When using vector arrow notation (\vec) combined with \lVert and \rVert, the norm won't cover the vector arrow if a capital letter issued for the vector, but instead cover extra "white space" below the letter.

Also \left\lVert # \right\rVert or \Big\lVert # \Big\rVert don't fix the problem, as they drastically increase vertical size covering much more white space above and below the vector. I would like to use this notation also in text, which is why vertical size matters a lot.

An optimal solution would be achieved if it was possible to simply shift the "||" upwards s.t. they cover the vector arrow and the least amount of white space below the letter, is that possible?

Example:

    \documentclass{article}
    \usepackage{amsmath}
\begin{document}

\[
\lVert \vec{A} \rVert = \left\lVert \vec{A} \right\rVert = \Big\lVert \vec{A} \Big\rVert
\]

\end{document}