IF `P_1, P_2, P_3, P_4` are points in a plane or space and `O` is the origin of vectors, show that `P_4` coincides with `Oiff( vec O P)_1+ vec P_1P_2+ vec P_2P_3+ vec P_3P_4= vec 0.`

Explanation

Let a,b,c and d are postion vector of `P_1, P_2, P_3, P_4`

thus,

`( vec O P)_1+ vec P_1P_2+ vec P_2P_3+ vec P_3P_4= vec 0.`


`=>vec a+(vec b-vec a)+(vec c-vec b)+(vec d-vec c)=0`

`=>vec d=0`

thus,

`P_4=0` and thus `P_4` coincides