If  $\overrightarrow{\alpha }+\overrightarrow{\beta }+\overrightarrow{\gamma }=a\overrightarrow{\delta }$ and $\overrightarrow{\beta }+\overrightarrow{\gamma }+\overrightarrow{\delta }=b\overrightarrow{\alpha },$ $\overrightarrow{\alpha }\text{ and }\overrightarrow{\delta }$ are non collinear vectors then

$\overrightarrow{\alpha }+\overrightarrow{\beta }+\overrightarrow{\gamma }+\overrightarrow{\delta }\text{ equals }$

$\text{ Given }\overrightarrow{\alpha }+\overrightarrow{\beta }+\overrightarrow{\gamma }=a\overrightarrow{\delta }-----\left(i\right)$

$\overrightarrow{\beta }+\overrightarrow{\gamma }+\overrightarrow{\delta }=b\overrightarrow{\alpha }---\left(ii\right)$

$\text{ From (i), }\overrightarrow{\alpha }+\overrightarrow{\beta }+\overrightarrow{\gamma }+\overrightarrow{\delta }=(a+1)\overrightarrow{\delta }----\left(iii\right)$

$\text{ From (ii), }\overrightarrow{\alpha }+\overrightarrow{\beta }+\overrightarrow{\gamma }+\overrightarrow{\delta }=(b+1)\overrightarrow{\alpha }-----\left(iv\right)$

From (iii) and (iv), we get 

$(a+1)\overrightarrow{\delta }=(b+1)\overrightarrow{\alpha }----\left(v\right)$

$\text{ Since }\overrightarrow{\alpha }\text{ is not parallel to }\overrightarrow{\delta }$

$\text{ From }\left(v\right),a+1=0\text{ and }b+1=0$

$\text{ From (iii), }\overrightarrow{\alpha }+\overrightarrow{\beta }+\overrightarrow{\gamma }+\overrightarrow{\delta }=\overrightarrow{O}$