Coulomb's Law in vector form

If vector F21 is the force exerted on charge q2 by charge q1 (Fig.),

Vector  F21 = k q1 q2 / (r12)^2     (r12 cap)

where unit vector r 12 cap is the unit vector  from q1 to q2.

If vector F12 is the force exerted on q1 due to q2,

Vector  F12 = k q1 q2 / (r21)^2     (r21 cap)

where unit vector r 21 cap is the unit vector  from q2 to q1.                           

                        Fig. Coulomb’s law in vector form                               

[Both unit vector u r 21 and r 12 have the same magnitude, and are oppositely directed]

      Vector F12 = k q1 q2 / (r12)^2     (-r12 cap)

or   Vector F12 = - k q1 q2 / (r12)^2     (r12 cap)

or   Vector F12 = -Vector F21

So, the forces exerted by charges on each other are equal in magnitude and opposite in direction.

Note :

A torsion balance is a sensitive device to measure force. It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.