I read somewhere that the electric field vector is perpendicular to the magnetic field vector
That, in general is not true. Think of a positive charge inside a solenoid. The electric field would be parallel to the magnetic field at points which lie on the line parallel to the magnetic field, containing the position of the charge.
wouldn’t that mean the electric field vector is the zero vector?
Not necessarily, but if the problem doesn't specify what the electric field is at that point, it is often assumed to be zero. (Unless the magnetic field is changing with time, in which case, an electric field would be induced in the volume. However, in this case, we are not given the magnetic field in a region, we are only given it's value at a point, so we are not left with much choice.)
should I use the inner/dot product when multiplying the velocity and magnetic field vector?
No, you should use the cross product (also sometimes called the outer product) while multiplying the velocity vector and the magnetic field. Using the dot product would yield a scalar, whereas the cross product gives a vector (which the force should be).