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. $\frac{\begin{array}{cc}2& 1\end{array}}{\begin{array}{ccc}H& G& O\end{array}}$
$HG:GO=2:1$
$\overrightarrow{OG}=\left(\frac{\vec{a}+\vec{b}+\vec{c}}{3}\right)<br>\overrightarrow{OH}=\vec{a}+\vec{b}+\vec{c}$
$\vec{N}$ is mid-point of $\overrightarrow{OH}$
$\therefore \vec{N}=\frac{\vec{a}+\vec{b}+\vec{c}}{2}$
$\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$
$=\vec{a}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)+\vec{b}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)$
$+\vec{c}-\left(\frac{\vec{a}+\vec{b}+\vec{c}}{2}\right)$
$=\left(\vec{a}+\vec{b}+\vec{c}\right)-\frac{3\left(\vec{a}+\vec{b}+\vec{c}\right)}{2}$
$=\frac{1}{2}\left(\vec{a}+\vec{b}+\vec{c}\right)$
$=-\frac{1}{2}\overrightarrow{OH}$
$=\frac{1}{2}\overrightarrow{HO}$