Let PQR be a triangle. Let → a = → QR , → b = → RP and → c = → PQ ....

\left|\overset{\to}{a}\right| = 12, \left|\overset{\to}{b}\right| = 4\sqrt{3}, \overset{\to}{b .}\overset{\to}{c} = 24$<br >$\overset{\to}{a} + \overset{\to}{b} + \overset{\to}{c} = 0$(Sides of triangle)<br >$\overset{\to}{b} + \overset{\to}{c} = - \overset{\to}{a}$<br > \left(\overset{\to}{b} + \overset{\to}{c}\right) . \left(\overset{\to}{b} + \overset{\to}{c}\right) = - \overset{\to}{a} . - \overset{\to}{a}$
${\left|\overset{\to}{b}\right|}^{2} + {\left|\overset{\to}{c}\right|}^{2} + 2\overset{\to}{b} . \overset{\to}{c} = {\left|\overset{\to}{a}\right|}^{2}$
$48 + {\left|\overset{\to}{c}\right|}^{2} + 2 \times 24 = {12}^{2}$
${\left|\overset{\to}{c}\right|}^{2} = 48$
$\dfrac{{\left|\overset{\to}{c}\right|}^{2}}{2} - \left|\overset{\to}{a}\right| = \dfrac{48}{2} - 12 = 12$
$\dfrac{{\left|\overset{\to}{c}\right|}^{2}}{2} + \left|\overset{\to}{a}\right| = \dfrac{48}{2} + 12 = 36$
\left|\overset{\to}{a} \times \overset{\to}{b} + \overset{\to}{c} \times \overset{\to}{a}\right| = \left|\overset{\to}{a} \times \overset{\to}{b} - \overset{\to}{a} \times \overset{\to}{c}\right|$<br >$= \left|\overset{\to}{a} \times \left(\overset{\to}{b} - \overset{\to}{c}\right)\right|$<br >$\overset{\to}{a} = - \overset{\to}{b} - \overset{\to}{c}$<br >$= \left|\left(- \overset{\to}{b} - \overset{\to}{c}\right) \times \left(\overset{\to}{b} - \overset{\to}{c}\right)\right| = \left|\left(\overset{\to}{b} - \overset{\to}{c}\right) \times \left(\overset{\to}{b} + \overset{\to}{c}\right)\right|$<br >$= 2\left|\overset{\to}{b} \times \overset{\to}{c}\right|$<br >$= 2\sqrt{{\left|\overset{\to}{b}\right|}^{2}{\left|\overset{\to}{c}\right|}^{2} - {\left(\overset{\to}{b} . \overset{\to}{c}\right)}^{2}} = 2\sqrt{48 \times 48 - {\left(24\right)}^{2}}$<br >$= 2 \times 24\sqrt{4 - 1} = 48\sqrt{3}$<br >$\overset{\to}{a} + \overset{\to}{b} = - \overset{\to}{c}$<br > \left(\overset{\to}{a} + \overset{\to}{b}\right) . \left(\overset{\to}{a} + \overset{\to}{b}\right) = - \overset{\to}{c} . - \overset{\to}{c}$
${\left|\overset{\to}{a}\right|}^{2} + {\left|\overset{\to}{b}\right|}^{2} + 2\overset{\to}{a} . \overset{\to}{b} = {\left|\overset{\to}{c}\right|}^{2}$
$\overset{\to}{a} . \overset{\to}{b} = \dfrac{- {\left|\overset{\to}{a}\right|}^{2}}{2} = - \dfrac{{12}^{2}}{2} = - 72$