Euler Angles

Euler angles are a 3-dimensional representation of an angle or direction and are, in practice, just vectors that require some operation to turn into directional vectors.

Instead of using the normal $x$, $y$ and $z$ elements for the elements though, a euler angle uses pitch ($p$), yaw ($y$) and roll ($r$), which can be used interchangably with the direction they rotate along.

Conversion to directional vector

With a given pitch, yaw and roll you can calculate a unit vector in the corresponding direction as follows:

$$ \hat{n} = \begin{pmatrix} \sin{y}\cos{p} \\ \sin{p} \\ \cos{y}\cos{p} \\ \end{pmatrix} $$

This formula should work for a system in which x = left/right, y = up/down and z = forward/backward, with positive y going up and negative y going down.