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Question

If A1,A2........An are the vertices of a regular plane polygon with n sides and O is the center then n−1∑i=1(→OAi×→OAi+1)=

A
(n+1)(OA1×OA2)
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B
(n+1)(OA2×OA1)
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C
(n1) (OA1×OA2)
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D
(n1) (OA2×OA1)
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Solution

The correct option is B (n−1) (→OA1×→OA2)∑n−1i=1(¯¯¯¯¯¯¯¯¯¯OAi×¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯OAi+1)¯¯¯¯¯¯¯¯¯¯OA1×¯¯¯¯¯¯¯¯¯¯OA2=|¯¯¯¯¯¯¯¯¯¯OA1||¯¯¯¯¯¯¯¯¯¯OA2|sinθ^n¯¯¯¯¯¯¯¯¯¯OA1,¯¯¯¯¯¯¯¯¯¯OA2,......¯¯¯¯¯¯¯¯¯¯¯OAn will be of same length and subtend same angle at the centre.∴∑n−1i=1(¯¯¯¯¯¯¯¯¯¯OAi×¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯OAi+1)=(n−1)|¯¯¯¯¯¯¯¯¯¯OA1||¯¯¯¯¯¯¯¯¯¯OA2|sinθ^n∴∑n−1i=1(¯¯¯¯¯¯¯¯¯¯OAi×¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯OAi+1)=(n−1)(¯¯¯¯¯¯¯¯¯¯OA1×¯¯¯¯¯¯¯¯¯¯OA2)Hence, option C is correct.

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