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Double Ordinate

Let ABC be an...

Question

Let $A B C$ be an acute scalene triangle, and $O$ and $H$ be its circumcentre and orthocentre respectively. Further let $N$ be the midpoint of $O H$ . The value of the vector sum $- - \to N A + - - \to N B + - - \to N C$ is

\to 0

(zero vector)

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- - \to H O

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\frac{1}{2} - - \to H O

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\frac{1}{2} - - \to O H

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Solution

The correct option is C $\frac{1}{2} - - \to H O$

Position vectors of $A = \to a$
$B = \to b C = \to c O = \to 0$
Centroid $G = \frac{\to a + \to b + \to c}{3}$
We know that,
$\frac{- - \to H G}{- - \to G O} = \frac{2}{1} \Rightarrow - - \to H G = \frac{2}{3} (\to a + \to b + \to c) \Rightarrow \to H - \to G = \frac{2}{3} (\to a + \to b + \to c) \Rightarrow \to H = (\to a + \to b + \to c)$

Midpoint of $O H$ is
$\to N = \frac{1}{2} (\to a + \to b + \to c)$
$Now, - - \to N A + - - \to N B + - - \to N C = 3 \to N - (\to a + \to b + \to c) = \frac{1}{2} (\to a + \to b + \to c) = \frac{1}{2} (- - \to H O)$

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