If $A B C D E F$ is a regular hexagon inscribed in a circle with centre $O$ , then the value of $- - \to A B + - - \to A C + - - \to A D + - - \to A E + - - \to A F$ equals

4 (- - \to A O)

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5 (- - \to A O)

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6 (- - \to A O)

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8 (- - \to A O)

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Solution

The correct option is C $6 (- - \to A O)$
$\to A B + \to A C + \to A D + \to A E + \to A F = \to b - \to a + \to c - \to a + \to d - \to a + \to c - \to a + \to f - \to a$
$= (\to b + \to c + \to d + \to e + \to f) - 5 \to a = - \to a - 5 \to a = - 6 \to a = 6 \to A O$

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¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ A C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ A E + ¯ ¯¯¯¯¯¯ ¯ A F = 3 ¯ ¯¯¯¯¯¯¯ ¯ A D = 6 ¯ ¯¯¯¯¯¯ ¯ A O

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¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ A C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ A E + ¯ ¯¯¯¯¯¯ ¯ A F = 6 ¯ ¯¯¯¯¯¯ ¯ A O

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\vec{A B,} \vec{A C,} \vec{A D,} \vec{A E}

\vec{A F}

\vec{A O,}

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¯ ¯¯¯¯¯¯ ¯ A B + ¯ ¯¯¯¯¯¯ ¯ A C + ¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ A E + ¯ ¯¯¯¯¯¯ ¯ A F

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