Let $\overrightarrow a $ , $\overrightarrow b $ and $\overrightarrow c $ be three unit vectors such that $\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0 $. If $\lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a $ and $\overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a $, then the ordered pair, $\left( {\lambda ,\overrightarrow d } \right)$ is equal to :

Question

Let $\overrightarrow a $ , $\overrightarrow b $ and $\overrightarrow c $ be three unit vectors such that
$\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0 $. If $\lambda = \overrightarrow a .\vec b + \vec b.\overrightarrow c + \overrightarrow c .\overrightarrow a $ and
$\overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a $, then the ordered pair, $\left( {\lambda ,\overrightarrow d } \right)$ is equal to :

$\left( {{3 \over 2},3\overrightarrow a \times \overrightarrow c } \right)$

$\left( { - {3 \over 2},3\overrightarrow c \times \overrightarrow b } \right)$

$\left( { - {3 \over 2},3\overrightarrow a \times \overrightarrow b } \right)$

$\left( {{3 \over 2},3\overrightarrow b \times \overrightarrow c } \right)$

Solution

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