Explain vector form of Coulomb's law and its importance.

Explanation

Suppose, position vectors of `q_1` and `q_2` are `r_(1)` and `r_2` respectively as shown in figure (a).
Let, force acting on `q_(1)` by `q_2` is F and force on `q_2` by `q_1` is `vecF_(21)`.
If 1 and 2 numbers are given to `q_(1)` and `q_2`, then `barr_(21)` is position vector from 1 to 2 and `barr_(12)` is position vector from 2 to 1.
By using triangle method for vector addition,
`vecr_(1) + vecr_(21) = vecr_(2)`
`therefore vecr_(21) = vecr_(2) - vecr_(1)` and `vecr_(12) = vecr_(1) - vecr_(2) =-vecr_(21)`
and `|vecr_(12)| = r_(12)` also `|vecr_(21)| = r_(21)`
`therefore vecr_(12) = vecr_(12)/vecr_(12)` and `vecr_(21) = vecr_(21)/vecr_(21)`
Force acting on `q_(2)` by `q_(1)`
`vecF_(21) = 1/(4pi epsilon_(0)).(q_(1)q_(2))/r_(21)^(2).hatr_(21)` and
Force acting on `q_(1)` by `q_(2)`
`vecF_(12) = 1/(4pi epsilon_(0)).(q_(1)q_(2))/r_(12)^(2).hatr_(12)`
but `hatf_(12) =-hatr_(21)`
`vecF_(21) =-vecF_(12)`