Let $$\overrightarrow a ,\,\overrightarrow b $$  and $$\overrightarrow c $$ be three non-zero vectors such that no two of these are collinear. If the vector $$\overrightarrow a + 2\overrightarrow b $$   is collinear with $$\overrightarrow c $$ and $$\overrightarrow b + 3\overrightarrow c $$   is collinear with $$\overrightarrow a $$ ($$\lambda $$ being some non-zero scalar) then $$\overrightarrow a + 2\overrightarrow b + 6\overrightarrow c $$   equals :

Solution :
Let $$\overrightarrow a + 2\overrightarrow b = t\overrightarrow c $$    and $$\overrightarrow b + 3\overrightarrow c = s\overrightarrow a ,$$    where $$t$$ and $$s$$ are scalars. Adding, we get
$$\eqalign{ & \overrightarrow a + 3\overrightarrow b + 3\overrightarrow c = t\overrightarrow c + s\overrightarrow a \cr & \Rightarrow \overrightarrow a + 2\overrightarrow b + 6\overrightarrow c = t\overrightarrow c + s\overrightarrow a - \overrightarrow b + 3\overrightarrow c \cr & \Rightarrow \overrightarrow a + 2\overrightarrow b + 6\overrightarrow c = t\overrightarrow c + \left( {\overrightarrow b + 3\overrightarrow c } \right) - \overrightarrow b + 3\overrightarrow c \cr & \Rightarrow \overrightarrow a + 2\overrightarrow b + 6\overrightarrow c = \left( {t + 6} \right)\overrightarrow c \,\,\,\,\,\,\,\,\,\,\left[ {{\text{using }}s\overrightarrow a = \overrightarrow b + 3\overrightarrow c } \right] \cr & \Rightarrow \overrightarrow a + 2\overrightarrow b + 6\overrightarrow c = \lambda \overrightarrow c ,{\text{ where }}\lambda = t + 6 \cr} $$