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# If $\overrightarrow a,\overrightarrow b\:\:and\:\:\overrightarrow c$ are non coplanar vectors, then for real $\lambda$, $[\lambda(\overrightarrow a+\overrightarrow b)\:\lambda^2\overrightarrow b\:\lambda\overrightarrow c]=[\overrightarrow a\:\overrightarrow b+\overrightarrow c\:\overrightarrow c]$, for:

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Given : $[\lambda (\overrightarrow a+\overrightarrow b)\:\lambda^2\overrightarrow b\:\lambda \overrightarrow c]=[\overrightarrow a\:\overrightarrow b+\overrightarrow c\:\overrightarrow c]$
$\Rightarrow\:\lambda^4\big[(\overrightarrow a+\overrightarrow b)\times \overrightarrow b\big].\overrightarrow c=\big[\overrightarrow a\times(\overrightarrow b+\overrightarrow c)\big].\overrightarrow c$
$\Rightarrow\:\lambda^4\big[\overrightarrow a\times \overrightarrow b).\overrightarrow c\big]=(\overrightarrow a\times \overrightarrow b).\overrightarrow c$
$\Rightarrow\:\lambda=\pm1$
$\therefore$ There are two values of $\lambda.$
answered Nov 10, 2013