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I write a vector with an arrow above the letter using \vec{p} and sometimes the vector needs to be primed using ' (or ^{\prime}, equivalently).
However, neither
\vec{p}'
nor
\vec{p'}
looks satisfactory (first option: prime dash overlaps with vector arrow, second option: arrow looks misplaced).
How would one write a primed vector?
Priming a vector denoted with the arrow is, as you discovered, not a really good thing to do. You want to prime the letter, not the whole construction.
\documentclass{article}
\newcommand{\pvec}[1]{\vec{#1}\mkern2mu\vphantom{#1}}
\begin{document}
$\vec{p}+\pvec{p}'=\pvec{p}''$
\end{document}
Adding a small amount of space in the first option should make it better:
{\vec{p}\,}'
There's a problem I don't get to understand. And it's the fact that if there are multiple atoms inside a \vec it gets recolocated. I comment a little bit here.
Disclaimer: this is not fully correct (because, for instance, the prime gets recolocated), but the fact that sometimes \vec{{}<something>} looks better than \vec{<something>} annoys me.
A possible solution is:
\let\originalvec\vec
\renewcommand*\vec[1]{\originalvec{\kern0pt #1}}
If that's not enough, you can mix it with egreg's answer
\newcommand*\pvec[1]{\vec{#1}\mkern1mu\vphantom{#1}}
Full compilable example
\documentclass{scrartcl}
\let\originalvec\vec
\renewcommand\vec[1]{\originalvec{\kern0pt #1}}
\newcommand*\pvec[1]{\vec{#1}\mkern2mu\vphantom{#1}}
\begin{document}
$\vec p + \pvec p' = \pvec p''$
$\vec h = \vec r \times \vec v = \vec r \times \pvec r'$
\end{document}
\skewchar of the font of the nucleus) and, more importantly, the superscript and the subscript are positioned before the accent is superimposed. Sorry, I cannot explain better in so few keystrokes...
In this period I have tried various solutions but I am not very convinced and satisfied. I insert some test:
\documentclass{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{stackengine}
\usepackage{tcolorbox}
\newcommand\barbelow[1]{\stackunder[1.2pt]{$#1$}{\rule{.8ex}{.075ex}}}
\newcommand\barbon[1]{\stackon[0.5pt]{$#1$}{\rule{.9ex}{.1ex}}}
\newcommand\barbons[1]{\stackon[0.5pt]{$#1$}{\rule{.9ex}{.1ex}}\mkern2mu\vphantom{#1}}
\newcommand\bvec[1]{\barbon{\vec{u}}}
%%----------------------------------------------------------------
%https://tex.stackexchange.com/questions/120029/how-to-typeset-a-primed-vector/253157#253157
\let\originalvec\vec
\renewcommand\vec[1]{\originalvec{\kern0pt #1}}
\newcommand*\pvec[1]{\vec{#1}\mkern2mu\vphantom{#1}}
%%----------------------------------------------------------------
\newcommand\barbona[1]{\stackon[1.2pt]{$#1$}{\rule{.9ex}{.1ex}}}
\newcommand\barbonb[1]{\stackon[1pt]{$#1$}{\rule{.9ex}{.1ex}}}
\newcommand\barbonc[1]{\stackon[0.8pt]{$#1$}{\rule{.9ex}{.1ex}}}
\newcommand\barbond[1]{\stackon[0.5pt]{$#1$}{\rule{.9ex}{.1ex}}}
%%----------------------------------------------------------------
\newcommand\barbonsa[1]{\stackon[1.2pt]{$#1$}{\rule{.9ex}{.1ex}}\mkern2mu\vphantom{#1}}
\newcommand\barbonsb[1]{\stackon[1pt]{$#1$}{\rule{.9ex}{.1ex}}\mkern2mu\vphantom{#1}}
\newcommand\barbonsc[1]{\stackon[0.8pt]{$#1$}{\rule{.9ex}{.1ex}}\mkern2mu\vphantom{#1}}
\newcommand\barbonsd[1]{\stackon[0.5pt]{$#1$}{\rule{.9ex}{.1ex}}\mkern2mu\vphantom{#1}}
\begin{document}
\begin{equation*}
\centering
\begin{cases}
\bar{\vec{u}} - \barbon{\vec{u}}\\
\bar{u}' - \barbon{u}'\\
\bar{\vec{u}}' - \barbon{\vec{u}}' - \barbon{\pvec{u}}' - \tcbox[tcbox raise base]{$\barbons{\vec{u}}'$}\\
\bar{C} - \barbon{C}\\
\bvec{u}\\
\pvec{a}+\pvec{b}+\pvec{t}+\pvec{u}\\
\barbon{\pvec{a}}+\barbon{\pvec{b}}+\barbon{\pvec{t}}+\barbon{\pvec{u}}\\
\end{cases}
\end{equation*}
\begin{center}
\begin{minipage}{.50\textwidth}
\begin{equation*}
\centering
\begin{cases}
\tcbox[tcbox raise base]{$\bar{u}$}\\
\tcbox[tcbox raise base]{$\barbona{\vec{u}}'$}\\
\tcbox[tcbox raise base]{$\barbonb{\vec{u}}'$}\\
\tcbox[tcbox raise base]{$\barbonc{\pvec{u}}'$}\\
\tcbox[tcbox raise base]{$\barbond{\pvec{u}}'$}\\
\end{cases}
\end{equation*}
\end{minipage}\hfill
\begin{minipage}{.50\textwidth}
\begin{equation*}
\centering
\begin{cases}
\tcbox[tcbox raise base]{$\bar{u}$}\\
\tcbox[tcbox raise base]{$\barbonsa{\vec{u}}'$}\\
\tcbox[tcbox raise base]{$\barbonsb{\vec{u}}'$}\\
\tcbox[tcbox raise base]{$\barbonsc{\vec{u}}'$}\\
\tcbox[tcbox raise base]{$\barbonsd{\vec{u}}'$}\\
\end{cases}
\end{equation*}
\end{minipage}
\end{center}
\end{document}
Someone raised this old (but good) question, so I might as well mention an alternative. You can use ISO-style bold italic vectors instead of an arrow.
In PDFLaTeX, you can do this with the isomath package.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb, isomath, bm}
\pagestyle{empty}
\begin{document}
\[ a \cdot \vectorsym{p}' \times \vectorsym{p}'' \]
\end{document}
In LuaLaTeX or XeLaTeX, you would use unicode-math.
\documentclass{article}
\usepackage{unicode-math}
\pagestyle{empty}
\newcommand\vectorsym[1]{\symbfit{#1}}
\begin{document}
\[ a \cdot \vectorsym{p}' \times \vectorsym{p}'' \]
\end{document}
If you want a bolder prime symbol to match the vector symbol, you can define
\newcommand\boldprime{\boldsymbol{'}}
For this to work in unicode-math, you must load a math font that comes in bold, or \setmathfont with .[version=bold].
\renewcommand\vec\mathbfin the pre-amble, and\vec{p}'in the document.\vec{p}^{\, \prime}, but it still doesn't look right. Maybe I'm overthinking this... Possibly decreasing the size of the prime would be an option, while increasing the spacing.\vec{p}^{\,\prime}looks alright myself; but I suppose it depends on precisely what you want to achieve with it. Do beware overthinking it though.