Question

Three non-zero non-collinear vectors $\overline{a},\overline{b},\overline{c}$ are such that $\overline{a}+3\overline{b}$ is collinear with $\overline{c}$, while $\overline{3}b+2\overline{c}$ is collinear with $\overline{a}$, then $\overline{a}+3\overline{b}+2\overline{c}=$

• A. $\overline{0}$
• B. $2\overline{a}$
• C. $3\overline{b}$
• D. $4\overline{c}$

Answer 1

The correct option is A $\overline{0}$

$\overrightarrow{a}+3\overrightarrow{b}=k\overrightarrow{c}$ ...(1)

$3b+2c=ma$ ...(2)

Substitute (1) in to (2)

$[k\overrightarrow{c}-\overrightarrow{a}]+2\overrightarrow{c}=ma$

$\Rightarrow k\overrightarrow{c}-\overrightarrow{a}+2\overrightarrow{c}=m\overrightarrow{a}$

$\Rightarrow \overrightarrow{c}[k+2]=\overrightarrow{a}[m+1]$

$\overrightarrow{a}$ and $\overrightarrow{c}$ are non-collinear

$k+2=0\Rightarrow k=-2$

$m+1=0\to m=-1$

$a+3b+2c$

$=kc+2c$

$k(-2)\times c+2c$

$=0$