Left align block of equations

Try to use the fleqn document class option.

\documentclass[fleqn]{article}

(See also http://en.wikibooks.org/wiki/LaTeX/Basics for a list of other options.)

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.

You can use \begin{flalign}, like the example bellow:

\begin{flalign}
    &f(x) = -1.25x^{2} + 1.5x&
\end{flalign}