If $\overrightarrow{a},\overrightarrow{b}\overrightarrow{c}$  are three non-zero vectors (no two of which are 

are three non-zero vectors (no two of which  $(\overrightarrow{a}+\overrightarrow{b},\overrightarrow{c})$  and 

$(\overrightarrow{b}+\overrightarrow{c},\overrightarrow{a})$ are collinear, then $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c} is not equal to$

 

[t is given that  $\overrightarrow{a}+\overrightarrow{b}$ is collinear with -

$\overrightarrow{c}$  and  $\overrightarrow{b}+\overrightarrow{c}$  is collinear with $\overrightarrow{a}$

$\therefore \overrightarrow{a}+\overrightarrow{b}=\lambda \overrightarrow{c}$ and $\overrightarrow{b}+\overrightarrow{c}=\mu \overrightarrow{a}$  for some scalars $\lambda$ and $\mu$

$\Rightarrow \overrightarrow{b}+\overrightarrow{c}=\mu (\widehat{i}\overrightarrow{c}-\overrightarrow{b})$                                         ( On eliminating $\overrightarrow{a}$ ]

$\Rightarrow (\mu +1)\overrightarrow{b}+(1-\mu \lambda )\overrightarrow{c}=\overrightarrow{0}$

$\Rightarrow \mu +1=0$ and $\mu \lambda =1[\because \overrightarrow{b}$ and $\overrightarrow{c}$ are non -collinear]]

$\begin{array}{r}\mu =-1\text{ and }\lambda =-1\\ \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0\end{array}$  

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$                         [Putting $\lambda =-1$ in $\overrightarrow{a}+\overrightarrow{b}=\lambda \overrightarrow{c}\text{ ] }$