Proving mutually perpendicular unit vectors?

Let $\vec{k}$ be a unit vector in 3-space and let $\vec{v}$ be a non-zero vector which is not parallel to $\vec{k}$.
If we define $\vec{i} = \frac {\vec{v} - (\vec{v} \cdot \vec{k}) \vec{k}}{a}$
and $\vec{j} = \frac{\vec{k} \times \vec{v}}{a}$
where $a = |\vec{v} - (\vec{v} \cdot \vec{k}) \vec{k}|$
${\vec{i}, \vec{j}, \vec{k}}$
are mutually perpendicular unit vectors with $\vec{k} \times \vec{i} = \vec{j}$