Write the expression for the Lorentz force F in vector form.

Hint: Lorentz Force is the force exerted on a charged particle q moving with velocity v through an electric field E and magnetic field B. $F = qE + q\left( {v \times B} \right)$.

Complete step by step answer:
Lorentz Force is the force exerted on a charged particle q with velocity v and E, B as electric and magnetic fields respectively. The entire electromagnetic force on the charged particle is the Lorentz Force.To find the direction of this force we apply right hand rule, the thumb of the right hand points in the direction of velocity, Index finger points in the direction of magnetic field and Middle finger points in the direction of Force.The magnetic force is proportional to the charge and perpendicular to velocity and magnetic field.As the value q is different for every charged particle and is a constant for a particular charged particle.
Therefore, the vectors will be Force, Electric field and Magnetic field.
$\vec F = q\vec E + q\left( {\vec v \times \vec B} \right) \\
\vec F = q\left( {\vec E + \vec v \times \vec B} \right) \\ $
The above expression of Lorentz force is in vector form where $\vec F$ is the total force exerted by a moving charged particle, $\vec E$ is the electric field, $\vec v$ is the velocity and $\vec B$ is the magnetic field.

Note: Lorentz force is used in electromagnetism. Lorentz force explains the mathematical equations along with the physical importance of forces acting on the charged particles that travel through the space containing electric and magnetic fields. In vector form, vectors are expressed in terms of their magnitude and direction or in terms of their components. Vectors have magnitude and direction whereas scalars just have magnitude. Do not confuse a vector with a scalar.