This is the problem...

Let $\displaystyle \vec{k} $ be a unit vector in 3-space and let $\displaystyle \vec{v} $ be a non-zero vector which is not parallel to $\displaystyle \vec{k}$.

If we define $\displaystyle \vec{i} = \frac {\vec{v} - (\vec{v} \cdot \vec{k}) \vec{k}}{a}$

and $\displaystyle \vec{j} = \frac{\vec{k} \times \vec{v}}{a} $

where $\displaystyle a = |\vec{v} - (\vec{v} \cdot \vec{k}) \vec{k}| $

show that

$\displaystyle {\vec{i}, \vec{j}, \vec{k}} $

are mutually perpendicular unit vectors with $\displaystyle \vec{k} \times \vec{i} = \vec{j} $