Question

If $\overline{p}$ is the position vector of the orthocentre and $\overline{g}$ is the position vector of the centroid of the triangle $ABC$ when circumcenter is the origin and if $\overline{p}=\lambda \overline{g}$ then $\lambda =$

• A. $3$
• B. $2$
• C. $\frac{1}{3}$
• D. $\frac{2}{3}$

Answer 1

The correct option is A $3$

Let $a,b,c$ be position vectors of $A,B,C$ respectively.

$\overline{p}$ is the position vector of orthocentre and $\overline{g}$ is the position vector of the centroid.

$⟹\overline{p}=\overline{a}+\overline{b}+\overline{c}$ and $\overline{g}=\frac{\overline{a}+\overline{b}+\overline{c}}{3}$

$⟹\overrightarrow{g}=\frac{\overrightarrow{p}}{3}\left(sectionformula\right)$

$⟹\overline{g}$ trisect $\overline{p}$
$⟹\overrightarrow{p}=3\overrightarrow{g}$
$⟹\overrightarrow{p}=\lambda \overrightarrow{g}$, where $\lambda =3$.

Hence, option A is correct.