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

Question

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

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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→$$+2b$$→$$+6c$$→$$=0$$.

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

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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

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