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Q)

If $a,b,c$ are lengths of sides of a $\Delta\:ABC$ and for any non collinear vectors $\overrightarrow u\:\:and\:\:\overrightarrow v$ if $(a-b)\overrightarrow u+(b-c)\overrightarrow v+(c-a)(\overrightarrow u\times\overrightarrow v)=0$, then the $\Delta$ is ?

• If three vectors $\overrightarrow x,\:\overrightarrow y,\:\overrightarrow z$ are non coplanar, then $a\overrightarrow x+b\overrightarrow y+c\overrightarrow z=\overrightarrow 0$ $\Rightarrow\:a=b=c=0$
Since $\overrightarrow u,\:\:and\:\:\overrightarrow v$ are non collinear,
$(a-b)\overrightarrow u+(b-c)\overrightarrow v +(c-a)(\overrightarrow u\times\overrightarrow v)$ are coplanar.
$\Rightarrow\: (a-b)\overrightarrow u+(b-c)\overrightarrow v +(c-a)(\overrightarrow u\times\overrightarrow v)=\overrightarrow 0$
$\Rightarrow\:(a-b)=(b-c)=(c-a)=0$
$\Rightarrow\:a=b=c$
$\therefore$ The $\Delta\:ABC$ is equilateral.