Lorentz Force : when a charged particle having charge q moves in a region , where both electric field $\overrightarrow{E}$ intensity and magnetic field $\overrightarrow{B}$ exist , it experiences a net force called Lorentz Force $\overrightarrow{F}$.
Lorentz Force ,$\overrightarrow{F}$ = Force on charge due to electric field + Force on charge due to magnetic field
$$\overrightarrow{F}= \overrightarrow{F_{e}} +\overrightarrow{F_{m}}$$
(i) Electric Force ( $\overrightarrow{F_{e}}$) , A charged particle having charge q placed in an electric field $\overrightarrow{E}$ experience a force given by ,
$$\overrightarrow{F_{e}} = q.\overrightarrow{E}$$
Direction of this force is same is same as that of electric field $\overrightarrow{E}$.
(ii) Magnetic force ($\overrightarrow{F_{m}}$) :
Magnetic force on a charge q moving with velocity $\overrightarrow{v}$ at a certain angle $\theta$ with magnetic field $\overrightarrow{B}$ is given by
$$\overrightarrow{F_{m}} = q (\overrightarrow{v} \times \overrightarrow{B})$$
$$Lorentz Force (\overrightarrow{F}) = \overrightarrow{F_{e}} +\overrightarrow{F_{m}}$$
$$\overrightarrow{F} = q.\overrightarrow{E} + q (\overrightarrow{v} \times \overrightarrow{B})$$
$$\overrightarrow{F} = q[\overrightarrow{E} + (\overrightarrow{v} \times \overrightarrow{B})]$$