Exam 1

1. Question

   Given two points $p=(2,4)$ and $q=(4,4)$ in a Cartesian coordinate system:
   
   1. What is the Manhattan distance $d_1(p,q)$?
   2. What is the Euclidean distance $d_2(p,q)$?
   3. What is the maximum distance $d_{\mathit{\infty }}(p,q)$?
   
   

   Solution

   The distances are visualized below in green ($d_1$), red ($d_2$), and blue ($d_{\mathit{\infty }}$).
   
   
   
   1. $d_1(p,q)=\sum _i|p_i-q_i|=|2-4|+|4-4|=2$.
   2. $d_2(p,q)=\sqrt{\sum _i(p_i-q_i{)}^2}=\sqrt{(2-4{)}^2+(4-4{)}^2}=2$.
   3. $d_{\mathit{\infty }}(p,q)=\underset{i}{max}|p_i-q_i|=max(|2-4|,|4-4|)=2$.