Solving a vector equation which seems to be indeterminate.

[plasmoid]

I have a vector equation:

\vec{A} \times \vec{B} = \vec{C}. \vec{A} and \vec{C} are known, and \vec{B} must be determined. However, upon trying to use Cramer's rule to solve the system of three equations, I find that the determinant we need is zero. I know now that I need to choose a "gauge" to proceed, but can someone outline what comes next? Thanks.

 

[Office_Shredder]

You know that AxB is perpendicular to both A and B, which means that B is a vector which is perpendicular to C. You have lots of choices for B because AxA = 0, so in particular
A\times B = A\times(B+\kappa A)
for any choice of kappa. In particular you can pick kappa to make B perpendicular to A. Once you have B is perpendicular to A and C, all that's left is to find its magnitude (which you can do using the fact that |AxB| = |A||B| if they are perpendicular)

 

[plasmoid]

You know that AxB is perpendicular to both A and B, which means that B is a vector which is perpendicular to C. You have lots of choices for B because AxA = 0, so in particular
A\times B = A\times(B+\kappa A)
for any choice of kappa. In particular you can pick kappa to make B perpendicular to A. Once you have B is perpendicular to A and C, all that's left is to find its magnitude (which you can do using the fact that |AxB| = |A||B| if they are perpendicular)

Makes sense. Now I have |B|. To get the components of \vec{B}, I can use

0 = A_{x} B_{x} + A_{y}B_{y}+ A_{z}B_{z}

and

0 = C_{x} B_{x} + C_{y}B_{y}+ C_{z}B_{z},

and

B^{2} = B_{x}^{2} + B_{y}^{2} + B_{z}^{2} .

Does that sound right?