Question

If a charged particle moves unaccelerated in a region containing electric and magnetic fields
(a) ${\displaystyle \overrightarrow{E} \text{must be perpendicular}\to \overrightarrow{B}}$
(b) ${\displaystyle \overrightarrow{v}\text{must be perpendicular}\to \overrightarrow{E}}$ 
(c) must be perpendicular to v_B

Answer 1

(a) ${\displaystyle \overrightarrow{E} \text{must be perpendicular}\to \overrightarrow{B}}$
(b) ${\displaystyle \overrightarrow{v}\text{must be perpendicular}\to \overrightarrow{E}}$

As the charged particle is not accelerated, the field ${\displaystyle \overrightarrow{E}}$ cannot be parallel to velocity . Hence, the velocity vecv is perpendicular to the electric field ${\displaystyle \overrightarrow{E}}$.

The magnetic force, i.e. force due to the magnetic field, acts in a direction perpendicular to the plane containing ${\displaystyle \overrightarrow{V}\text{and}\overrightarrow{B}}$. Hence, in order to counterbalance the magnetic force, an equal and opposite electric force must be applied along the same axis in which the magnetic force is acting. Hence vecE must be perpendicular to ${\displaystyle \overrightarrow{v}}$ and  vec B.