If $\vec a,\vec b,\vec c$ are three non-zero vectors, no two of which are collinear and the vector $\vec a+\vec b$ is collinear with $\vec c$, $\vec b+\vec c$ is collinear with $\vec a$, then $\vec a+\vec b+\vec c=$
Answer & explanation
Correct answer: option 4
It is given that
$\vec a+\vec b$ is collinear with $\vec c$ and $\vec b+\vec c$ is collinear with $\vec a$.
$∴\vec a+\vec b=λ\vec c$ and $\vec b+\vec c =μ \vec a$
$⇒\vec a+\vec b=λ(μ\vec a-\vec b)$ [On eliminating $\vec c$]
$⇒(λμ-1)\vec a-(λ+1)\vec b=\vec 0$
$⇒λμ-1=0$ and $λ+1=0$ [∵$\vec a,\vec b$ bare non-collinear]
$⇒λ=-1$ and $μ=-1$
Substituting $λ=-1$ in $\vec a+\vec b=λ\vec c$, we get $\vec a+\vec b+\vec c=\vec 0$