Orthogonal vector in a plane I have two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$. Now I want to find a vector $\boldsymbol{v}$ orthogonal to $\boldsymbol{a}$ that is in the plane spanned by $\boldsymbol{a}$ and $\boldsymbol{b}$. Is it ok if I do $\boldsymbol{v} = \boldsymbol{a} \times (\boldsymbol{a} \times \boldsymbol{b})$?

Yes. $\boldsymbol a\times (\boldsymbol a\times \boldsymbol b)$ is perpendicular to both $\boldsymbol a$ and $\boldsymbol a\times \boldsymbol b$. Being perpendicular to $\boldsymbol a\times \boldsymbol b$ means being on the plane generated by $\boldsymbol a$ and $\boldsymbol b$.
See the other answer for a less expensive computation that outputs such a vector.
