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Vector Algebra question

2023 · 10 Apr · Shift 2 · Q30
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  5. /2023 · 10 Apr · Shift 2 · Q30

Vector Algebra question

2023 · 10 Apr · Shift 2 · Q30

JEE MainMathematicsVector AlgebraMCQ+4 / −1
If the points P\mathrm{P}P and Q\mathrm{Q}Q are respectively the circumcenter and the orthocentre of a △ABC\triangle \mathrm{ABC}△ABC, then PA→+PB→+PC→\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}PA+PB+PC is equal to :
  1. A
    QP→\overrightarrow {QP}QP​
  2. B
    PQ→\overrightarrow {PQ}PQ​
  3. C
    2PQ→2\overrightarrow {PQ}2PQ​
  4. D
    2QP→2\overrightarrow {QP}2QP​
View written solutionFree

Correct answer: B

  1. Use the circumcenter as origin

Let us take point PPP (the circumcenter) as the origin of vectors. Then the position vectors of A,B,CA,B,CA,B,C are

PA⃗, PB⃗, PC⃗.\vec{PA},\ \vec{PB},\ \vec{PC}.PA, PB, PC.

Since PPP is the circumcenter, these are the radius vectors of the circumcircle.

  1. Known vector formula for the orthocenter

If the circumcenter is taken as origin, then the position vector of the orthocenter QQQ is

PQ⃗=PA⃗+PB⃗+PC⃗.\vec{PQ}=\vec{PA}+\vec{PB}+\vec{PC}.PQ​=PA+PB+PC.

This is a standard result in triangle geometry.

For completeness, let

PA⃗=a⃗,PB⃗=b⃗,PC⃗=c⃗.\vec{PA}=\vec a,\quad \vec{PB}=\vec b,\quad \vec{PC}=\vec c.PA=a,PB=b,PC=c.

Then the orthocenter has position vector

q⃗=a⃗+b⃗+c⃗.\vec q=\vec a+\vec b+\vec c.q​=a+b+c.

Hence

PA⃗+PB⃗+PC⃗=PQ⃗.\vec{PA}+\vec{PB}+\vec{PC}=\vec{PQ}.PA+PB+PC=PQ​.
  1. Match with the options

Therefore,

PA→+PB→+PC→=PQ→.\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=\overrightarrow{PQ}.PA+PB+PC=PQ​.

So the correct option is:

B\boxed{\text{B}}B​

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