Question

derivation of coulombs law in vector form

Answer 1

If I understand your question correctly, the explanation is the following. The direction of the vector F⃗ 12F→12 depends on the sign of q1q1 and q2q2. If they have the same sign it will point in one direction, while if they have opposite sign it will point in the opposite direction. Therefore you cannot put the absolute value in the vector expression of Coulomb's law, because you would lose some information about the resulting F⃗ 12F→12 vector.

When you only care about the magnitude of this vector, on the other hand, you have to put the absolute value because this has to be a positive number. Note that |q1||q2||q1||q2| is the same positive number, independent of the sign of q1q1 and q2q2.
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