Trigonometry & Inverse Trigonometry
Circumcentre and Orthocentre Vector Relation
nta_pyq_2023_apr
Grade 11
Question:
If $P$ and $Q$ are the circumcentre and orthocentre of $\triangle ABC$, then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}$ is equal to
$2\overrightarrow{QP}$
$2\overrightarrow{PQ}$
$\overrightarrow{PQ}$
$\overrightarrow{QP}$
Step-by-Step Solution
Key Concept: Take $P$ as origin. Centroid $G=\frac{\vec{a}+\vec{b}+\vec{c}}{3}$. By Euler line: $Q=3G$ (since $P,G,Q$ are in ratio $1:2$).
$\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=\overrightarrow{PQ}$.
Correct Answer: 3