Is there a way to find a least squares solution for a vector using a system of cross product equations?
For example $\vec{A}, \vec{B}, \vec{C}, \vec{D}$ are all known quantities in 3D space:
$$\vec{A} = [1,0,-.01]$$ $$\vec{B} = [.01,1,0]$$ $$\vec{C} = [1,.01,-.01]$$ $$\vec{D} = [-.01,1,.01]$$
These vectors are related by the following cross products where $\vec{E}$ is unknown.
$$\vec{A} \times \vec{B} = \vec{E}$$ $$\vec{C} \times \vec{D} = \vec{E}$$ $$\vec{A} \times \vec{D} = \vec{E}$$ $$\vec{C} \times \vec{B} = \vec{E}$$
$\vec{E}$ should be the same in each relationship and should be close to $[0,0,1]$