Mathematics - Vector Question with Solution | TestHub

${\mathrm{R}}^2={\mathrm{P}}^2+{\mathrm{Q}}^2+2\mathrm{PQ}\cos \theta$
$4{\mathrm{R}}^2={\mathrm{P}}^2+4{\mathrm{Q}}^2+4\mathrm{PQ}\cos \theta$
$4{\mathrm{R}}^2={\mathrm{P}}^2+{\mathrm{Q}}^2-2\mathrm{PQ}\cos \theta$
$\mathrm{On}(1)+(2),5{\mathrm{R}}^2=2{\mathrm{P}}^2+2{\mathrm{Q}}^2$
$\mathrm{On}(3)\times 2+(2),12{\mathrm{R}}^2=3{\mathrm{P}}^2+6{\mathrm{Q}}^2$
$2{\mathrm{P}}^2+2{\mathrm{Q}}^2-5{\mathrm{R}}^2=0$
$3{\mathrm{P}}^2+6{\mathrm{Q}}^2-12{\mathrm{R}}^2=0$
$\frac{{\mathrm{P}}^2}{-24+30}=\frac{{\mathrm{Q}}^2}{24-15}=\frac{{\mathrm{R}}^2}{12-6}$ $\frac{{\mathrm{P}}^2}{6}=\frac{{\mathrm{Q}}^2}{9}=\frac{{\mathrm{R}}^2}{6}$ or ${\mathrm{P}}^2:{\mathrm{Q}}^2:{\mathrm{R}}^2=2:3:2$