Question

Let $\overline{a},\overline{b}$ and $\overline{c}$ be three nonzero vectors no two of which are collinear. If $\overline{a}+2\overline{b}$ is collinear with $\overline{c}$ and $\overline{b}+3\overline{c}$ is collinear with $\overline{a}$ then $\overline{a}+2\overline{b}+6\overline{c}$ is

• A. $0$
• B. $\lambda \overline{a}$
• C. $\lambda \overline{b}$
• D. $\lambda \overline{c}$

Answer 1

The correct option is A $0$

$\overrightarrow{a}+2\overrightarrow{b}=k\overrightarrow{c}$ -(1)
$\overrightarrow{b}+3\overrightarrow{c}=k_1\overrightarrow{a}$ -(2)

Solving for $\overrightarrow{b}$
$\overrightarrow{b}=(k_1\overrightarrow{a}-3\overrightarrow{c})$ -(3)
put (3) in (1)
$\overrightarrow{a}+2k_1\overrightarrow{a}-6\overrightarrow{c}=k\overrightarrow{c}$
as $\overrightarrow{a}$ & $\overrightarrow{c}$ are non collinear
$k_1=\frac{-1}{2}$
$k=-6$
$\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}=0$