Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If…
Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec{a}+2 \vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+3 \vec{c}$ is collinear with $\vec{a}$ ( $\lambda$ being some non-zero scalar) then $\vec{a}+2 \vec{b}+6 \vec{c}$ equals
$\lambda \vec{a}$
$\lambda \vec{b}$
$\lambda \overrightarrow{\mathbf{c}}$
0
Solution
$(\vec{a}+2 b)=t_1 \vec{c}$
and $\vec{b}+3 \vec{c}=t_2 \vec{a}$
$(1)-2 \times(2) \Rightarrow \vec{a}\left(1+2 t_2\right)+\vec{c}\left(-t_1-6\right)=0 \Rightarrow 1+2 t_2=0 \Rightarrow t_2=-1 / 2 \& t_1=-6$.
Since $\vec{a}$ and $\vec{c}$ are non-collinear.
Putting the value of $t_1$ and $t_2$ in (1) and (2), we get $\vec{a}+2 \vec{b}+6 \vec{c}=\overrightarrow{0}$