Describe geometrically the set of solutions to the following equations in 3-space

Question

Given $a, b \in \mathbb{R}^3$ and $\lambda \in \mathbb{R}$ I'm looking to describe, geometrically, the set of all $x \in \mathbb{R}^3$ satisfying both equations

$a\cdot x = \lambda$

$a \times x = b$

The first equation tells me that $x$ lies on a plane. The second equation tells me that the area of the parallelogram determined by $x$ and $a$ is fixed (and equal to $\|b\|$). Having drawn some pictures I think that these two equations determine a circle in the plane, however I am unsure as to how I can verify if I'm correct or not.

Thanks

Answer 1

Once $a$ is fixed, the map $x \rightarrow a \times x$ is a linear map from $\mathbb{R}^3$ to itself. Its kernal is spanned by $a$ and its image is the plane perpendicular to $a$. In particular unless $b$ is perpendicular to $a$ the second equation has no solutions.

If $b$ is perpendicular to $a$ then the solution space to $a \times x = b$ is one dimensional, and moreover it is parallel to the vector $a$ and hence perpendicular to the plane $a \cdot x = \lambda$ so we get a single solution to this system of equations.

Answer 2

In general there is no solution to those two simultaneous equations.

You are right that the first equation says that $x$ lies in a specific plane. Now choose the point in that plane that corresponds to $x$. That dictates an exact value for $b = a\times x$. So the dimension of the space of all possible $b$ values is only 2. Since 3-space has three dimensions, most values of $(\lambda,b)$ will not admit a solution for $x$.