1.

If A_(1),A_(2)….., A_(n) are theverticesof a regularplanepolygonwith n sidesand o isits centre. Thenshowthat 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))

Answer»

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Solution :Since `vec(OA)_(1) , vec(OA)_(2) ,…….vec(OA)_(n)` are llvectors ofsamemagnitudeand anglebetweenany twoconsectivevectorsis samei.e.,`(2pi//n)`
`:. , vec(OA)_(1) xx vec(OA)_(2) = a^(2) sin .(2pi)/(n). P`
where`hat(p) ` isperpendicularto planeof 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`


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