Question

Prove that in a hexagon the sum of the vectors drawn to its vector from the origin is zero

Answer 1

in a Hexagon Vector Drawn from each vertex to center would be = 0

Step-by-step explanation:

in Hexagon all internal angles are 120°

Vector Drawn from each vertex to center would be ( A is side of Hexagon or Radius of circum circle of hexagon)

A∠60   = ACos60 + iAsin60  = A/2  + i√3/2A

A ∠120 = ACos120 + iAsin120  = -A/2  + i√3/2A

A∠180 = ACos180 + iAsin190  = -A  

A∠240 = ACos240 + iAsin240  = -A/2  - i√3/2A

A∠300 =   ACos300 + iAsin300  = A/2  - i√3/2A

A∠360 =  A∠0  = A

Addding all

A/2  + i√3/2A -A/2  + i√3/2A  -A  - A/2  - i√3/2A + A/2  - i√3/2A

= 0

Vector Drawn from each vertex to center would be = 0

Answer 2

Answer:

let the origin of hexagon be O

thus,

OA=-OD

OB=-OE

OC= -OF

The sum of all the vectors drawn from the center of regular hexagon to it's vertices is

=OA+OB+OC+OD+OE+OF

=-OD-OE-OF+OD+OE+OF

=0

Hence, the sum of vectors drawn to it's vertices from the origin is ZERO

HENCE PROVED