\begin{align} not working as expected

None of the align, gather and equation environments must be typeset within of a $...$ pair, since those environments start the math mode for themselves. This is true for the starred variants of the mentioned environments as well.

Some side remarks:

• Use \sin instead of sin -- it's a math operator which should not be typeset in italics.
• \dfrac{}{} is (sometimes) a better replacement for nicer fractions.
• \DeclairedPairedDelimiters{\abs}{\lvert}{\rvert} is much better than |some stuff|

---

\documentclass{article}
\usepackage{mathtools}


\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\begin{document}
%$
%m\dfrac{v^2}{r}=qvB\\
%\dfrac{2qV}{r}=qvB\\
%\dfrac{2qV}{r}=qB\sqrt{\dfrac{2qV}{m}}\\
%\dfrac{V}{r}=B\sqrt{\dfrac{q}{2m}}\sqrt{V}\\
%$

\begin{align*}
\abs{\vec{F}} &=\abs{q}\abs{\vec{v}}\abs{\vec{B}}\abs{\sin(\theta)}\\
m\dfrac{v^2}{r}& =qvB\\
\dfrac{2qV}{r} &=qvB\\
\dfrac{2qV}{r} & =qB\sqrt{\dfrac{2qV}{m}}\\
\dfrac{V}{r} &=B\sqrt{\dfrac{q}{2m}}\sqrt{V}\\
\end{align*}

\begin{align*}
\dfrac{V}{r} &=\lambda\sqrt{V}  &\lambda\equiv B\sqrt{\dfrac{q}{2m}}
\end{align*}

\begin{gather*}
\dfrac{V}{r}=\lambda\sqrt{V},\lambda\equiv B\sqrt{\dfrac{q}{2m}}
\end{gather*}

\begin{equation}
\dfrac{V}{r}=\lambda\sqrt{V},\lambda\equiv B\sqrt{\dfrac{q}{2m}}
\end{equation}

\end{document}