Step 1 (derivation): Work done in rotating a dipole from $\theta_1$ to $\theta_2$ against the torque $\tau=pE\sin\theta$: $W=\int_{\theta_1}^{\theta_2}pE\sin\theta\,d\theta = pE(\cos\theta_1-\cos\theta_2)$.

Step 2: Taking $\theta_1=90°$ (reference, $U=0$) as standard: $U(\theta) = -pE\cos\theta = -\vec p\cdot\vec E$.

Step 3: $U$ is minimum ($=-pE$) when $\theta=0°$ ($\vec p \parallel \vec E$, stable equilibrium); $U$ is maximum ($=+pE$) when $\theta=180°$ ($\vec p$ antiparallel to $\vec E$, unstable equilibrium).

Step 4 (numerical): Dipole vector from $-q$ to $+q$: $\vec d = (3\ mm,4\ mm)$, magnitude $d=5$ mm $=5\times10^{-3}$ m.

Step 5: $\vec p = q\vec d = 1.0\times10^{-12}\times(3\times10^{-3}\hat i+4\times10^{-3}\hat j) = (3\times10^{-15}\hat i+4\times10^{-15}\hat j)\ C\cdot m$.

Step 6: $\vec\tau = \vec p\times\vec E$; since $\vec E = 1000\hat i$: $\vec\tau = (3\times10^{-15}\hat i+4\times10^{-15}\hat j)\times1000\hat i = 4\times10^{-15}\times1000(\hat j\times\hat i) = 4\times10^{-12}\times(-\hat k)$.

Step 7: $\vec\tau = -4\times10^{-12}\ \hat k\ N\cdot m$ (magnitude $4\times10^{-12}$ N·m, directed into the page).