Orthocentre of an equilateral triangle $A B C$ is the origin $O$ . If $A = ¯ ¯ ¯ a, B = ¯ ¯ b, C = ¯ ¯ c$ then $¯ ¯¯¯¯¯¯ ¯ A B + 2 ¯ ¯¯¯¯¯¯ ¯ B C + 3 ¯ ¯¯¯¯¯¯ ¯ C A =$

3 ¯ ¯ c

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3 ¯ ¯ ¯ a

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0

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3 ¯ ¯ b

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Solution

The correct option is B $3 ¯ ¯ ¯ a$
$¯ ¯¯¯¯¯¯ ¯ A B + 2 ¯ ¯¯¯¯¯¯ ¯ B C + 3 ¯ ¯¯¯¯¯¯ ¯ C A$
$= \to b - \to a + 2 (\to c - \to b) + 3 (\to a - \to c)$
$= - \to b + 2 \to a - \to c$ ----------(1)
orthocentre = circumcenter (property of equalatoral $Δ$ )
so circumecentre $= \frac{\to a + \to b + \to c}{3}$
$0 = \frac{\to a + \to b + \to c}{3}$
$- \to b = \to a + \to c$
so put in (1)
$3 \to a$

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