State Coulomb's law in vector form and from it define relative permittivity.

Explanation

Coulomb.s law states that the force of attraction or repulsion between to charges is directly proportional to product of their charges and inversely proportional to square of distance between them.
Let `q_1 and q_2` be two charges placed .r. distance apart in air.
Let `vecF_(21)` = Force on charge `q_2` due to charge `q_1`

`:. vecF_(12)=1/(4piin_0)(q_1xxq_2)/(r_(12)^2) hatr_(12)`
`hatr_(12)` is a unit vector pointing from charge `q_1` to `q_2`
`vecF_(12)` Force on charge `q_1` due to charge `q_2 `
Therefore `vecF_(12)=1/(4piin_0)(q_1xxq_2)/(r_(21)^2)hatr_(21)`
where `hatr_(21)` is a unit vector pointing from charge `q_1 and q_2`
Relative permittivity : According to Coulomb.s law :
Force between charges `q_1 and q_2` in a medium,
`F_m=1/(4piin_0K)(q_1xxq_2)/(r^2)`
For air , K = 1
`:.` Force between charges `q_1 and q_2` in air,
`F_(air)=1/(4piin_0)(q_1xxq_2)/r^2`
Dividing (1) and (2)
`K=(F_("air"))/(F_m)`
Relative permittivity or dielectric constant of a medium is defined as the ratio of force between two charges placed at certain distance apart in air to the force between them placed same distance apart in the medium.