This is the problem...

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}|
show that
{\vec{i}, \vec{j}, \vec{k}}
are mutually perpendicular unit vectors with \vec{k} \times \vec{i} = \vec{j}