Prove magnetic force cannot lie in the velocity-field plane

[humanallien]

here is the problem:

prove that the vector of the magnetic force cannot be parallel to the plane that contains the vector of the particle's velocity and magnetic field.

someone please help me to solve this, thanks so much for your time.

 

[Tide]

here is the problem:

prove that the vector of the magnetic force cannot be parallel to the plane that contains the vector of the particle's velocity and magnetic field.

someone please help me to solve this, thanks so much for your time.

By definition of cross product you have both

$$\vec v \cdot \vec v \times \vec B = 0$$

and

$$\vec B \cdot \vec v \times \vec B = 0$$

which say that both $\vec v$ and $\vec B$ are perpendicular to $\vec F = \vec v \times \vec B$.

 

[humanallien]

thanks for your advice, but it does not help me with the problem because I have to assume that I do not know the definition of F = v X B for the prove. Therefore, I want to proof that the force vector is NOT parallel to the plane without considering the definition of the croos product

 

[Tide]

Since the fundamental behavior of charged particles interacting with a magnetic field comes only from observation (Thomson, Lorentz, etc.) it would seem unlikely that you could "prove" the result by other means. It would be like attempting to prove the electric force on a charged particle acts parallel to the electric field - without invoking Coulomb's Law. Good luck!