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

where λ is a non-zero scalar.

$\vec 0$

It is given that $\vec a +2\vec b$ is collinear with $\vec c$ and $\vec b+3\vec c$ is collinear with $\vec a$.

$∴\vec a +2\vec b=x\vec c$, and $\vec b+3\vec c=y\vec a$ for some $x, y ∈R$

$∴\vec a +2\vec b+6\vec c=(x+6)\vec c$

Also, $\vec a +2\vec b+6\vec c=(1+2y)\vec a$

$∴(x+6)\vec c=(1+2y)\vec a$

$⇒x+6=0$ and $1+2y=0$   [∵ $\vec a, \vec c$ are non-collinear]

$⇒x=-6$ and $y = -1/2$

$⇒\vec a+2\vec b+6\vec c=\vec 0$