Suppose we have two vectors - $A$ and $B$. We know that the dot product between them is given by $AâB$. However, what if we want to calculate the dot product of their sum and difference?

$(A+B)â(AâB)=?$The dot product is distributive over vector addition. Thus, we can use the FOIL method to expand the expression:

$(A+B)â(AâB)=AâAâAâB+BâAâBâB$Recall that the dot product is commutative, meaning that $AâB=BâA$. We can cancel out two terms in the expression above to simplify it further:

$(A+B)â(AâB)=AâAâBâB$The dot product of a vector with itself is the magnitude of the vector squared. Thus, we can simplify this expression once again:

$(A+B)â(AâB)=âĢâĢAâĢâĢ_{2}ââĢâĢBâĢâĢ_{2}$And there we have it! We got a nice, simplified expression involving the magnitudes of the two vectors. Pretty neat!