A charged particle travels undeflected through a region containing...

The Lorentz force on a charged particle is given by $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$.

For the particle to travel undeflected, the net force must either be zero or parallel to its velocity $\vec{v}$.

If $\vec{F} = 0$, then $\vec{E} = -(\vec{v} \times \vec{B})$. This implies $\vec{E}$ is perpendicular to $\vec{B}$. Thus, $\vec{E}$ is not parallel to $\vec{B}$, making option (1) possible.

If $\vec{E} \parallel \vec{B}$ and $\vec{v} \parallel \vec{E}$, then $\vec{v} \parallel \vec{B}$, which gives $\vec{v} \times \vec{B} = 0$. The net force is $\vec{F} = q\vec{E}$. Since $\vec{E} \parallel \vec{v}$, the force is parallel to the velocity. The particle accelerates or decelerates along a straight line without deflection, making option (3) possible.

For option (2), if $\vec{v} \parallel \vec{B}$, the magnetic force is zero. For no deflection, the electric force $q\vec{E}$ must be parallel to $\vec{v}$, meaning $\vec{E} \parallel \vec{v} \parallel \vec{B}$. This contradicts the given condition that $\vec{E}$ is not parallel to $\vec{B}$.

For option (4), if $\vec{E} \parallel \vec{B}$ and $\vec{v}$ is not parallel to $\vec{E}$, the magnetic force is non-zero and perpendicular to $\vec{B}$, while the electric force is parallel to $\vec{B}$. The net force cannot be parallel to $\vec{v}$, causing the particle to deflect.

Answer: $\vec{E}$ is not parallel to $\vec{B}$; $\vec{E} \parallel \vec{B}$, $\vec{v} \parallel \vec{E}$