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Addition of Vectors

Let ABC be ...

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

A

\to 0

(zero vector)

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B

\to H O

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C

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

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D

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

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Solution

The correct option is B $\frac{1}{2} \to H O$
$- - \to H E = - - \to N E - - - \to N H$
$\Rightarrow \frac{1}{2} - - \to A H = \frac{1}{2} [- - \to N B + - - \to N C] - \frac{1}{2} - - \to O H$ .... $(∵ H E ⊥ B C, A H ⊥ B C)$
$\Rightarrow - - \to H E ∥ - - \to A H$
$H E = R cos (A), A H = 2 R cos (A) \Rightarrow - - \to H E = \frac{1}{2} - - \to A H$
$E$ midpoint of $B C$
$\Rightarrow - - \to N E = \frac{1}{2} [- - \to N B + - - \to N C]$
$N$ midpoint of $O H$
$\Rightarrow - - \to N H = \frac{1}{2} - - \to O H \Rightarrow - - \to N H - - - \to N A = - - \to N B + - - \to N C + - - \to O H$
$\Rightarrow - - \to N A + - - \to N B + - - \to N C$ = - $- - \to O H + \frac{1}{2} - - \to O H$
= $\frac{1}{2} - - \to H O$

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