Let a→,b→ and c→ be three non-zero vectors, no two of which are collinear. If the  vector a→+2b→ is collinear with c→, and b→+3c →is collinear with,a→ then a→+2b→+6c→ is equal to

Let a→+2b→=xc→ and b→+3c→=ya→ , where x,y are scalars Then a→+2b→+6c→=(x+6)c→ and also 2( b→+3c→)=2ya→⇒a→+2b→+6c→=(1+2y)a→ S  (x+6)c→=(1+2y)a→ . Since a→ and  c→ are non-zero and non-collinear,  we have x+6=0 and 1+2y=0, i.e., x=−6 and y=−1/2 . In either case, we have a→+2b→+6c→=0.