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 :

$\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0$ <br><br>$\Rightarrow$ $${\left| {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right|^2}$$ = 0 <br><br>$\Rightarrow$ ${{{\left| {\overrightarrow a } \right|}^2}}$ + ${{{\left| {\overrightarrow b } \right|}^2}}$ + ${{{\left| {\overrightarrow c } \right|}^2}}$ + <br>${2\left( {\overrightarrow a .\overrightarrow b } \right)}$ + ${2\left( {\overrightarrow b .\overrightarrow c } \right)}$ + ${2\left( {\overrightarrow c .\overrightarrow {a} } \right)}$ = 0 <br><br>$\Rightarrow$ 3 + $${2\left( {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a } \right)}$$ = 0 <br><br>$\Rightarrow$ 3 + 2$\lambda$ = 0 <br><br>$\Rightarrow$ $\lambda$ = $- {3 \over 2}$ <br><br>$$\overrightarrow d = \overrightarrow a \times \vec b + \vec b \times \overrightarrow c + \overrightarrow c \times \overrightarrow a $$ <br><br>= $${\overrightarrow a \times \overrightarrow b + \overrightarrow b \times \left( { - \overrightarrow a - \overrightarrow b } \right) + \left( { - \overrightarrow a - \overrightarrow b } \right) \times \overrightarrow a }$$ <br><br>= $${\overrightarrow a \times \overrightarrow b - \overrightarrow b \times \overrightarrow a + }$$ $${\overrightarrow b \times \overrightarrow b - \overrightarrow a \times \overrightarrow a - \overrightarrow b \times \overrightarrow a }$$ <br><br>= $${\overrightarrow a \times \overrightarrow b + \overrightarrow a \times \overrightarrow b + \overrightarrow a \times \overrightarrow b }$$ <br><br>= ${3\left( {\overrightarrow a \times \overrightarrow b } \right)}$