Mathematics - Vector Question with Solution | TestHub

$\left|\overrightarrow{\mathrm{a}}\right|=12,\mathrm{ }\left|\overrightarrow{\mathrm{b}}\right|=4\sqrt{3},\mathrm{ }\overrightarrow{\mathrm{b}.}\overrightarrow{\mathrm{c}}=24$
$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=0$(Sides of triangle)
$\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=-\overrightarrow{\mathrm{a}}$
$\left(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\right).\left(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\right)=-\overrightarrow{\mathrm{a}}.-\overrightarrow{\mathrm{a}}$
${\left|\overrightarrow{\mathrm{b}}\right|}^2+{\left|\overrightarrow{\mathrm{c}}\right|}^2+2\overrightarrow{\mathrm{b}}.\overrightarrow{\mathrm{c}}={\left|\overrightarrow{\mathrm{a}}\right|}^2$
$48+{\left|\overrightarrow{\mathrm{c}}\right|}^2+2\times 24={12}^2$
${\left|\overrightarrow{\mathrm{c}}\right|}^2=48$
$\frac{{\left|\overrightarrow{\mathrm{c}}\right|}^2}{2}-\left|\overrightarrow{\mathrm{a}}\right|=\frac{48}{2}-12=12$
$\frac{{\left|\overrightarrow{\mathrm{c}}\right|}^2}{2}+\left|\overrightarrow{\mathrm{a}}\right|=\frac{48}{2}+12=36$
$\left|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\times \overrightarrow{\mathrm{a}}\right|=\left|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}\right|$
$=\left|\overrightarrow{\mathrm{a}}\times \left(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}\right)\right|$
$\overrightarrow{\mathrm{a}}=-\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}$
$=\left|\left(-\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}\right)\times \left(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}\right)\right|=\left|\left(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}\right)\times \left(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\right)\right|$
$=2\left|\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}\right|$
$=2\sqrt{{\left|\overrightarrow{\mathrm{b}}\right|}^2{\left|\overrightarrow{\mathrm{c}}\right|}^2-{\left(\overrightarrow{\mathrm{b}}.\overrightarrow{\mathrm{c}}\right)}^2}=2\sqrt{48\times 48-{\left(24\right)}^2}$
$=2\times 24\sqrt{4-1}=48\sqrt{3}$
$\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=-\overrightarrow{\mathrm{c}}$
$\left(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right).\left(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}\right)=-\overrightarrow{\mathrm{c}}.-\overrightarrow{\mathrm{c}}$
${\left|\overrightarrow{\mathrm{a}}\right|}^2+{\left|\overrightarrow{\mathrm{b}}\right|}^2+2\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}={\left|\overrightarrow{\mathrm{c}}\right|}^2$
$\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}=\frac{-{\left|\overrightarrow{\mathrm{a}}\right|}^2}{2}=-\frac{{12}^2}{2}=-72$