I have the following system of nonlinear equations:
$$\mathbf{a}^{\rm T} \mathbf{x} = \alpha,$$ $$\mathbf{b}^{\rm T} \mathbf{x} = \beta,$$ $$\lvert \lvert \mathbf{x} \rvert \rvert = 1,$$
where $\mathbf{a},\mathbf{b}, \mathbf{x} \in \mathbb{R}^3$ with $\mathbf{a},\mathbf{b} \ne \mathbf{0}$ and $\alpha, \beta \in \mathbb{R}$. Is there any simple way to find a solution $\mathbf{x}$ for this nonlinear system?
One approach is to find a parameterization of the linear subsystem, then use this parameterization to find the solution for the full system.
In particular, suppose that $\mathbf {a,b}$ are linearly independent. All solutions to the system $$ \mathbf a^T \mathbf x = \alpha, \\ \mathbf b^T \mathbf x = \beta $$ can be written in the form $\mathbf x = \mathbf u + t \mathbf v$ with $t \in \Bbb R$, where $\mathbf x = \mathbf u$ is any particular solution to the equation and $\mathbf v$ is a vector satisfying $\mathbf a^T \mathbf v = \mathbf b^T \mathbf v = 0$ (for instance, $\mathbf v$ could be the cross-product $\mathbf {v = a \times b}$). From there, we are looking for any value of $t$ for which $$ \|\mathbf u + t\mathbf v\|^2 = 1 \implies\\ \|\mathbf v\|^2 t^2 + 2 \langle \mathbf {u},\mathbf v\rangle t + \|\mathbf u\|^2 = 1 $$ (where $\langle \mathbf u, \mathbf v\rangle$ denotes the dot-product of $\mathbf u$ and $\mathbf v$), which is a quadratic equation.