If I have two orthonormal vectors in $\mathbb{R}^3$, $\mathbf{u}$ & $\mathbf{v}$ (so both unit length). and another vector $\mathbf{w}$ that is orthogonal to $\mathbf{u}$ (not to $\mathbf{v}$ though).
How can I find -with least calculations & roots- a third orthonormal vector to both $\mathbf{u}$ & $\mathbf{v}$ (this vector may be in the direction of w, but doesn't have to be, just that I have an orthonormal basis. $\mathbb{R}^3 = \text{span}\{\mathbf{u}, \mathbf{v},\mathbf{w}\}$
EDIT: it seems I have now 3 methods: using Gram-Schmidt ($\mathbf{w}' = \mathbf{w}- \frac{\mathbf{w}\cdot\mathbf{v}}{\mathbf{v}\cdot\mathbf{v}}\mathbf{v} - \frac{\mathbf{w}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\mathbf{u}$). Using cross products and finally solving the equations that the dot products between the vectors must be 0. Guess it is up to experience what is easiest? Or is there anymore advice/am I doing something stupid?
Knowing an additional vector $w$ orthogonal to $u$ doesn't help, because for example it could just be $w = -v$. Use the cross product $w = u \times v$ to get your vector $w$, for example see http://en.wikipedia.org/wiki/Cross_product