This question comes from Ted Shifrin's Multivariable Mathematics. The question states:
Given the nonzero vector $\textbf{a} \in \mathbb{R}^{3}$, $\textbf{a} \cdot \textbf{x} = b \in \mathbb{R}$, and $\textbf{a} \times \textbf{x} = \textbf{c} \in \mathbb{R}^{3}$, can you determine the vector $\textbf{x} \in \mathbb{R}^{3}$? If so give a geometric construction of $\textbf{x}$.
I'm trying to visualize what the vector $\textbf{x}$'s would look like in this scenario, but I think I'm getting confused by everything happening. So $\textbf{a} \cdot \textbf{x} = b$ is giving me the equation of a plane, in particular an affine plane from the one at the origin. As well $\textbf{a} \times \textbf{x} = \textbf{c}$ is giving me vector orthogonal to $\textbf{a}$ and $\textbf{x}$ respectively. The norm of this cross product also gives me the area of the parallelogram spanned by $\textbf{a}$ and $\textbf{x}$. Even with all these properties I'm still having trouble figuring out how to solve for $\textbf{x}$.
Would I be able to get some assistance? If I'm lucky maybe the man himself actually might pop in to provide guidance.
BIG HINT: Note that the equation $\mathbf a\times\mathbf x = \mathbf c$ tells us that $\mathbf a$ and $\mathbf c$ are orthogonal. Draw a picture of an example. Say $\mathbf c$ is a vector along the positive $\mathbf e_3$-axis. Then $\mathbf a$ must be a vector in the $xy$-plane. Draw it in a convenient location. Last, relative to it, where must $\mathbf x$ be located in order for the equation $\mathbf a\times\mathbf x = \mathbf c$ to hold? You mentioned the area of a parallelogram. Draw the possible locations (and there are infinitely many) of $\mathbf x$ to arrive at that.