If $\overrightarrow {\rm{a}} ,\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{c}}$ are unit vectors such that $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = \vec 0$, then the value ${\mathop{\rm of}\nolimits} \overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}}$ is

We have, $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = 0$

and ${\overrightarrow {\rm{a}} ^2} = 1,{\overrightarrow {\rm{b}} ^2} = 1,{\overrightarrow {\rm{c}} ^2} = 1$

$\Rightarrow$ ${\vec a^2} + \vec a \cdot \vec b + \vec a \cdot \vec c + \vec b \cdot \vec a + {\vec b^2} + \vec b \cdot \vec c + \vec c \cdot \vec a + \vec c \cdot \vec b + {\vec c^2}$

$= 0$
$\Rightarrow$ ${\vec a^2} + {\vec b^2} + {\vec c^2} + 2(\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a) = 0$

and $\overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}} = \overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{c}} ]$

$\Rightarrow$ $1 + 1 + 1 + 2(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}} ) = 0$

$\Rightarrow$ $\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a = - \frac{3}{2}$