If the magnetic field is parallel to the positive $y$-axis and the charged particle is moving along the positive $x$-axis (Figure),which way would the Lorentz force be for
$(a)$ an electron (negative charge),
$(b)$ a proton (positive charge).

(N/A) The Lorentz force on a moving charge is given by $F = q(v \times B)$.
Here,the velocity vector $v$ is along the positive $x$-axis,i.e.,$v = v\hat{i}$.
The magnetic field $B$ is along the positive $y$-axis,i.e.,$B = B\hat{j}$.
The cross product $v \times B$ is $(\hat{i} \times \hat{j}) = \hat{k}$,which is along the positive $z$-axis.
$(a)$ For an electron,the charge $q = -e$. Thus,the force $F = -e(vB\hat{k}) = -evB\hat{k}$. The force is along the negative $z$-axis.
$(b)$ For a proton,the charge $q = +e$. Thus,the force $F = +e(vB\hat{k}) = +evB\hat{k}$. The force is along the positive $z$-axis.