A proton moving eastward at right angles to a uniform magnetic field experiences a northward force. What is the direction of the force on a beta particle moving southward?

PLACEHOLDER ANSWER - the answer is a direction. Take east, north and up as $$\hat{i}$$, $$\hat{j}$$, $$\hat{k}$$. For the proton, $$\hat{i} \times \vec{B}$$ is along $$\hat{j}$$, which needs $$\vec{B}$$ along $$-\hat{k}$$ (downward). For a negatively charged beta particle moving along $$-\hat{j}$$, the force is along $$-\left(-\hat{j} \times -\hat{k}\right) = -\hat{i}$$, i.e. westward.