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i am studying the coulomb's law.then i encounter the formula for force between charges i.e $F=k q1q2/r^2$. They represent it with its vector form. now my question what is the use of that vector form. why we need it. if we have a normal coulomb formula why we represent it in vector form.what is the thing we cannot do with coulomb's normal law but we can easily do that thing with its vector form. They also do the same with gravitation law and after giving its normal formula they represent it in vector form. i don't understand what is the purpose behind it. why it is important to represent laws like gravitation and coulomb's law in vector form and what is this vector form?

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When studying the interaction of several forces, in different directions, it is an easy way to keep track of the direction each of these forces are in.

Also mathematically vectors tend to be very easy to work complex problems with since they natively support actions that appear naturally in many cases such as the dot product, vector product, ...

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The magnitudes of both Coulomb's and Newton's laws are given by

$$ F_C = k\frac{Q q}{r^2} , \qquad F_N = \frac{G M m}{r^2}. $$

Now ask yourself what are these statements really telling you. In their current forms they are simply numbers or you could say both $F_C$ and $F_N$ in their current form are a measure of the strength of either attraction/repulsion between two charged particles or it measures the strength which two particles are gravitationally attracted to each other.

In what direction are these forces directed? This is answered by using a vector equation. Please see image below (kindly borrowed from Wikipedia.)

enter image description here

You might like to look here and here.

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Coulomb's law deals with charge-charge interaction based on action at a point concept It means forces acting between the two charges are action_reaction pair.Vector form of Coulomb's law clearly explains the action_reaction pair.

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