Try this:
\begin{flalign*}
&|\vec a| = \sqrt{3^{2}+1^{2}} = \sqrt{10} & \\
&|\vec b| = \sqrt{1^{2}+23^{2}} = \sqrt{530} &\\
&\cos v = \frac{26}{\sqrt{10} \cdot \sqrt{530}} &\\
&v = \cos^{-1} \left(\frac{26}{\sqrt{10} \cdot \sqrt{530}}\right) &\\
\end{flalign*}
The & sign separates two columns, so an & at the beginning of a line means that the line starts with a blank column.