Hi all,

Suppose I have a known 3-D vector field $\hat{b}$, is it always possible to
express another vector field(Let's call it A) which is perpendicular to this
vector field in the following form:

\begin{equation}
\vec{A}=\hat{b} \times \nabla \Phi + \hat{b} \times (\hat{b} \times \nabla \Psi)
\end{equation}

We can see the above representation certainly guarantees that $\vec{A}$ is
perpendicular to $\hat{b}$.

If the answer is yes, how should one represent $\Phi$ or $\Psi$ in terms of
$\vec{A}$?

If the answer is no, what is the criteria for such representation to be
appropriate?

Thanks!