Find the vector sum of `N` coplanar forces, each of the magnitude `F`,when each force makes an angle of `2pi//N` with that preceding it.

Explanation

Let (A) be the magnitude of each vector and the first vector is along x-axis. Then the various vectors will be making angles with the x-axis ltbRgt as ` 0, ( 2 pi) /N, 9 4 pi)/N, ( 6 pi) /N ,……( (N- a)/N 2 pi`.
Let ` R_x ` and ` R_y` be the (X) and Y-component of resultant vector ` vec T`. The
` R-x = A cos 0^@ + A cos ( 2 pi) /N + A cos ( 4 pi) N + .....`
` + A cos ( N- 10 ( 2 pi)/N = A sum _ ( i=0)^(( N-1)) cos ( 2 pi i)/N` ltbRgt and ` R_y = A sun_ (i=0)^(( N-1)) sin ( 2 pi i) /N`
But ` sun_(i=0)^(( N-1)) cos ( 2 pi i)/N = sum_(i=0)^((N-1)) sin ( 2pi i) /N =0`
:. ` R_x = R_y =0`.
Hence, ` R= sqrt (R_x^2 + R_y^2) =0`
Note. the total angle subtende by (N) vectors with the inital cector `= ( 2 pi // N) xx N= 2 pi `. It means these (N) vectors can be represented by the (N) sides of a closed polygon taken in one order .Hence their vector sum is zero.