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# If $\overrightarrow a,\overrightarrow b\:and\:\overrightarrow c$ are non zero vectors that are pair wise non-collinear and if $\overrightarrow a+3\overrightarrow b$ is collinear with $\overrightarrow c$ and $\overrightarrow b+2\overrightarrow c$ is collinear with $\overrightarrow a$, then $\overrightarrow a+3\overrightarrow b+6\overrightarrow c=?$

Toolbox:
• Two collinear vectors are parallel vectors.
Given: $\overrightarrow a+3\overrightarrow b$ is collinear with $\overrightarrow c$ and
$\overrightarrow b+2\overrightarrow c$ is collinear with $\overrightarrow a$.
$\Rightarrow\:\overrightarrow a+3\overrightarrow b=\lambda\overrightarrow c$
and $\overrightarrow b+2\overrightarrow c=\mu \overrightarrow a$
$\Rightarrow\:(1-\mu)\overrightarrow a +4\overrightarrow b+(2-\lambda)\overrightarrow c=\overrightarrow o$
$\Rightarrow\:\overrightarrow a+3\overrightarrow b+6\overrightarrow c=\lambda\overrightarrow c+6\overrightarrow c=(\lambda+6)\overrightarrow c$
Also $\overrightarrow a+3\overrightarrow b+6\overrightarrow c=\overrightarrow a +3(\overrightarrow b+2\overrightarrow c)=\overrightarrow a+3\mu\overrightarrow a=(3\mu+1)\overrightarrow a$
$\Rightarrow\:(\lambda+6)\overrightarrow c=(3\mu+1)\overrightarrow a$
But it is given that $\overrightarrow a,\overrightarrow b$ and $\overrightarrow c$ are not collinear pairwise.
$\Rightarrow\:\lambda+6=0$ and $3\mu+1=0$
$\Rightarrow\:\lambda=-6$ and $\mu=-\large\frac{1}{3}$
$\therefore:\overrightarrow a+3\overrightarrow b+6\overrightarrow c=\overrightarrow 0$