`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))=(n-1) (vec(OA)_1 xx vec(OA)_(2))`

Explanation

To solve the problem, we need to show that:

\[ \sum_{i=1}^{n-1} \vec{OA_i} \times \vec{OA_{i+1}} = (n-1) \left( \vec{OA_1} \times \vec{OA_2} \right) \]

where \( \vec{OA_i} \) represents the position vector of the vertex \( A_i \) of a regular polygon with \( n \) sides, and \( O \) is the center of the polygon.

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