My apologies for this dumb question but I have never gotten such a strange answer from Mathematica before. I was hoping that someone could tell me what Mathematica is doing and why. This actually started with a physics problem: The torque on a dipole (M) in a magnetic field (B) is given by torque (T) T = M x B and the torque applied to dipole particle at a distance (R) from the center of rotation is
T = R x F where (F) acts on the particle then we get a vector algebra problem: R x F = M x B where each capital letter is a 3 vector in R3.

(1) The first question is how do you solve for the three components of (F)?

(2) I tried right and left multiplications with the other three vectors, but Mathematica always returns a null set.

(3) Next, I tried reducing the complexity by creating what should be a very simple problem:

Solve R x F = R x B. This should have the answer F = B. Mathematica doesn't think so. It returns this silliness (below). Would someone please tell me why my Mathematica code is wrong and what happened here??? So, what happened to (fx) and why doesn't F = B appear as an answer?

In[1523]:= Clear[m, b, r, f, t, mx, my, mz, bx, by, bz, rx, ry, rz,
fx, fy, fz, tx, ty, tz]

mx =.;
my =.;
mz =.;
bx =.;
by =.;
bz =.;
fx =.;
fy =.;
fz =.;
rx =.;
ry =.;
rz =.;
tx =.;
ty =.;
tz =.;

m = {mx, my, mz};
b = {bx, by, bz};
r = {rx, ry, rz};
f = {fx, fy, fz};
t = {tx, ty, tz};

ss = Solve[{Simplify[Cross[r, f] == Cross[r, b]]}, {fx, fy, fz}]

Simplify[ss]

During evaluation of In[1523]:= Solve::svars: Equations may not give solutions for all "solve" variables.

Out[1544]= {{fy -> (fx ry)/rx - (-by rx + bx ry)/rx, 
  fz -> (fx rz)/rx - (-bz rx + bx rz)/rx}}

Out[1545]= {{fy -> (by rx - bx ry + fx ry)/rx, 
  fz -> (bz rx - bx rz + fx rz)/rx}}

Many thanks to all the great users that took the the time to read and respond to my questions!!! Muchas Gracias!!