Exactly!

Given a and m

a = {a1, a2, a3}
m = {m1, m2, m3}

Define with an arbitrary parameter x

f = x a - m

Then

Cross[a, m]
Cross[f, a]

Hi Eric and Daniel, Thank you for your kind replies.

I should have been more precise in my statement of the problem.

What I was looking for was a general solution for F when R x F = M x B.

(Hi Eric): The special case R x F = R x B apparently does not have a solution F = B according to what you were saying. Would you be so kind as to explain this ?? Is there a good reference book on vector algebra that discusses these operations in detail?? (Lang??)

So, Is there a general solution for F when R x F = M x B ??

If yes, how do I code this in Mathematica??

When I coded the three sets of equations for { fx, fy, fz } Mathematica only gave TWO answers:

fy --> (by rx - bx ry + fx ry)/rx

fz --> (bz rx - bx rz + fx rz)/rx}}

Why did Mathematica not provide an answer for (fx)??

Again, Thank you kindly gentleman for your replies!!!

Solve R x F = R x B. This should have the answer F = B

Not an answer to your larger question, but this assertion is false for cross products.