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Section Formula

If Aa , B...

Question

If $A (¯ ¯ ¯ a)$ , $B (¯ ¯ b)$ and $C (¯ ¯ c)$ be the vertices of a triangle $A B C$ whose circumcentre is the origin then orthocentre is given by

A

¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c

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B

\frac{¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c}{3}

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C

\frac{¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c}{2}

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D

\frac{¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c}{4}

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Solution

The correct option is A $¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c$
P.V. of circumcentre $= (0, 0, 0)$
P.V. of circumcentre $= \frac{\to a + \to b + \to c}{3}$
$c e n t r o i d = \frac{o r t h o c e n t r e + 2 c i r c u m c e n t r e}{3}$ (section formula)
$\frac{\to a + \to b + \to c}{3} = \frac{o r t h o c e n t r e}{3}$
orthocentre $= \to a + \to b + \to c$

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