How do I absolutely make sure I have the longest length? An explanation perhaps via video with a diagram could help. But any explanation whatsoever could also help a lot. Thanks.

Question

572.Given square ABCD, choose a point O that is not outside the square and form the vector v = OA + OB + OC + OD. By trying various possible positions for O, find the shortest and longest possible v.

Answer 1

by observation if O is in center then the four vectors add to zero

if O is on any corner then four vectors add to side times 2 plus side times √2

move O off corner and not only is at least one vector less but others have negative components.

adding less to negative is less then original.