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

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`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
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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise
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  3. A(1),A(2), …. A(n) are the vertices of a regular plane polygon with n ...

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  11. Let veca, vecb and vecc be non - coplanar unit vectors, equally inclin...

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  12. If veca, vecb, vecc are vectors such that |vecb|=|vecc| then {(veca+ve...

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  13. For any two vectors vecu and vecv prove that (1+|vecu|^2(1+|vecv|^20=(...

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  14. Let vecu and vecv be unit vectors. If vecw is a vector such that vecw+...

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  15. find three- dimensional vectors, vecv1, vecv2 and vecv3 " satisfying "...

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  16. Let V be the volume of the parallelepied formed by the vectors, veca ...

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