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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
Question
If aBC is a triangle whose orthocenter is $\mathrm{P}$ and the circumcenter is $\mathrm{Q}$, then prove that $P A$

$+\overline{P C}+\overline{P B}=2 \overline{P Q}$

✓

Answer

Let G be the centroid of the ∆ ABC.

Let $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{G}, \mathrm{Q}$ have position vectors $\bar{a}, \bar{b}, \bar{c}, \overline{\boldsymbol{g}}, \bar{q}$ w.r.t. P. We know that $\mathrm{Q}, \mathrm{G}, \mathrm{P}$ are

collinear and G divides segment QP internally in the ratio 1 : 2.

$\therefore \bar{g}=\frac{1 \cdot \bar{p}+2 \bar{q}}{1+2}=\frac{2 \bar{q}}{3}$

$[\because \bar{p}=\overline{0}]$

$\therefore 3 \bar{g}=2 \bar{q}$

$\therefore \frac{3(\bar{a}+\bar{b}+\bar{c})}{3}=2 \bar{q}$

$\therefore \bar{a}+\bar{b}+\bar{c}=2 \bar{q}$

$\therefore \overline{\mathrm{PA}}+\overline{\mathrm{PB}}+\overline{\mathrm{PC}}=2 \overline{\mathrm{PQ}}$

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