`A_(1),A_(2), …. A_(n)` are the vertices of a regular plane polygon with n sides and O ars its centre. Show that `underset(i=1)overset(n-1)sum(vec(OA_(i))xxvec(OA)_(i+1))=(1-n) (vec(OA)_2 xx vec(OA)_(1))`

Explanation

`vec(OA)_(1) , vec(OA)_(1) …..,vec(OA)_(n)` All vectors are of same magnitude, say a, and angle between any two consecuitve vectors parallel to the plane of the plane of the polygon.
Let ` vec(OA)_(1) xx vec(OA)_(2)=a^(2) sin "" (2pi)/n hatp`
Now `underset(i=1)overset(n-1)sumvec(OA)_(1)xx vec(OA)_(i+1)= underset(i+1)overset(n-1)suma^(1)sin""(2pi)/n hatp`
` (n-1) a^(2) sin "" (2pi)/n hatp`
` (n-1) [-vec(OA)_(2) xx vec(OA)_(1)]`
`(1-n) [ vec(OA)_(2) xx vec(OA)_(1)]`
R.H.S