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 $\boldsymbol{v} = \boldsymbol{a} \times (\boldsymbol{a} \times \boldsymbol{b})$?

As @Tal-Botvinnik says, it's perfectly OK in general. There is a failure case, which is when $a$ and $b$ are linearly dependent; then $a \times b = 0$, and you're in trouble. 
A computationally slightly cheaper approach is to take 
$$
n = b - \frac{b \cdot a}{a \cdot a} a.
$$
This also works in the case where $a$ and $b$ are vectors in any inner-product space, not just 3-dimensions. 
(BTW, it also fails on the same case as the cross-product approach; that's because the answer isn't unique in that case.)
