If $\overrightarrow {\rm{a}} ,\overrightarrow {\rm{b}}$ and $\overrightarrow {\rm{c}}$ are three vectors such that $\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} = \vec 0$ and $|\overrightarrow {\rm{a}} | = 2$,$|\overrightarrow {\rm{b}} | = 3$ and $|\overrightarrow {\rm{c}} | = 5$, then the value of $\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}}$ is

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

and ${{\rm{a}}^{\vec 2}} = 4,\overrightarrow {{{\rm{b}}^2}} = 9,\overrightarrow {{{\rm{c}}^2}} = 25$

$\therefore (\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} ) \cdot (\overrightarrow {\rm{a}} + \overrightarrow {\rm{b}} + \overrightarrow {\rm{c}} ) = \overrightarrow {\rm{0}}$

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

$\Rightarrow$
$\Rightarrow$ $4 + 9 + 25 + 2(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} + \overrightarrow {\rm{b}} \cdot \overrightarrow {\rm{c}} + \overrightarrow {\rm{c}} \cdot \overrightarrow {\rm{a}} ) = 0$

$\Rightarrow$ $\vec a \cdot \vec b + \vec b \cdot \vec c + \vec c \cdot \vec a = \frac{{ - 38}}{2} = - 19$