Question

Let $ABC$ be an acute scalene triangle, and $O$ and $H$ be its circumcentre and orthocentre respectively. Further, let $N$ be the mid-point of $OH$. The value of the vector sum $\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$ is

• A. $\vec{0}$ (zero vector)
• B. $\overrightarrow{HO}$
• C. $\frac{1}{2} \overrightarrow{HO}$
• D. $\frac{1}{2} \overrightarrow{OH}$

Answer 1

Answer: (C)

Let position vector of $\Delta\,ABC$ are $A \left(\vec{a}\right), B \left(\vec{b}\right) and \left(\vec{c}\right)$

Let circumcentre of $\Delta\,ABC, O$ (origin)
Centroid of $\Delta\,ABC$ is $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$
We know that centroid divide orthocentre and circumcentre in $2 : 1$
i.e. $\frac{\begin{matrix}2&1\end{matrix}}{\begin{matrix}H&G&O\end{matrix}}$
$HG : GO=2 : 1$
$\overrightarrow{OG}=\left(\frac{\vec{a}+\vec{b}+\vec{c}}{3}\right)
\overrightarrow{OH}=\vec{a}+\vec{b}+\vec{c}$
$\vec{N}$ is mid-point of $\overrightarrow{OH}$
$\therefore \vec{N}=\frac{\vec{a}+\vec{b}+\vec{c}}{2}$
$\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$
$=\vec{a}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)+\vec{b}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)$
$+\vec{c}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)$
$=\left(\vec{a}+\vec{b}+\vec{c}\right)-\frac{3\left(\vec{a}+\vec{b}+\vec{c}\right)}{2}$
$=\frac{1}{2}\left(\vec{a}+\vec{b}+\vec{c}\right)$
$=-\frac{1}{2} \overrightarrow{OH} $
$=\frac{1}{2} \overrightarrow{HO}$