Distance between two points

The distance between two points $A$ and $B$ on the plane is the module of the fixed vector that determines: $$d(A,B)=|\overrightarrow{AB}|$$ In coordinates, if $A=(a_1,a_2)$ and $B =(b_1,b_2)$, then we have: $$d(A,B)=|\overrightarrow{AB}|=|(b_1-a_1,b_2-a_2)|=\displaystyle \sqrt{(b_1-a_1)^2+(b_2-a_2)^2}$$

To calculate the distance between points $A = (3, 4)$ and $B = (2,-5)$. $$d (A, B) =|\overrightarrow{AB}| = | (2-3,-5-4) | = | (-1,-9) | = \displaystyle \sqrt{(-1)^2+(-9)^2}=\sqrt{82}$$