Home/ Questions/Q 416495
Next
In Process
Mini Suresh
Asked: 2022-11-05T13:27:32+05:30 In:

If `A_(1),A_(2)….., A_(n)` are the vertices of a regular plane polygon with n sides and o is its centre. Then show that
`Sigma_(n-1)^(i-1) (overset(to)(OA)_(i) xx overset(to)(OA)_(i+1) ) =(1-n)(overset(to)(OA)_(2)xxoverset(to)(OA)_(1))`

If `A_(1),A_(2)….., A_(n)` are the vertices of a regular plane polygon with n sides and o is its centre. Then show that
`Sigma_(n-1)^(i-1) (overset(to)(OA)_(i) xx overset(to)(OA)_(i+1) ) =(1-n)(overset(to)(OA)_(2)xxoverset(to)(OA)_(1))`
Parabola
Leave an answer

Leave an answer

Browse

1 Answer

  1. Answer – Since `vec(OA)_(1) , vec(OA)_(2) ,…….vec(OA)_(n)` are ll vectors of same magnitude and angle between any two consective vectors is same i.e., `(2pi//n)`
    `:. , vec(OA)_(1) xx vec(OA)_(2) = a^(2) sin .(2pi)/(n). P`
    where `hat(p) ` is perpendicular to plane of polygon.
    Now `overset(n-1)underset(i=1)(Sigma) (vec(OA)_(i) xx vec(OA)_(i+1) ) = overset(n-1)underset(i=1)(Sigma) a^(2). sin.(2pi)/(n).p`
    ` =(n-1) .a^(2) sin.(2pi)/(n).hat(p)`
    `=(n-1) [vec(OA)_(1) xx vec(OA)_(2)]`
    `=(1-n)[vec(OA)_(2) xx vec(OA)_(1)] =RHS`

    • 0