How to solve this . which equations to use?

Question

An electron is initially moving along the x-axis with a speed of 10 m s−1 in a magnetic field given by B~ = 10G~1z. Show that the motion of the electron will always be in the x, y-plane and never along the z-axis.

Answer 1

The force on a charge will be f = qVxB where V is the velocity vector and B is the magnetic field vector. Since V is along x and B is along Z, the cross product of VxB and thus the force is in the x-y direction, so there is no force in the z direction ( i x k = j)

Edit: in case it isn’t clear, while the velocity is initially in the in the x direction, the force will give it a component in the y direction so you need to show also that the cross product of a vector with i and j components and a vector in the k direction will always be in the x-y plane.