Question

Three non-zero collinear vectors $a,b$ and $c$ are such that $a+3b$ is collinear with $c$ and $3b+2c$ is collinear with $a$.

Then, $a+3b+2c$ is equal to:

• A. $0$
• B. $2a$
• C. $3b$
• D. $4c$

Answer 1

The correct option is A

$0$

Explanation for correct option

Given that $a+3b$ is collinear with $c$ and $3b+2c$ is collinear with $a$.

Let, $a+3b=kc$ and $3b+2c=ma$

Then, $3b=kc-a$

and, $kc-a+2c=ma$

$$\begin{gathered} \Rightarrow kc+2c=ma+a \\ \Rightarrow c(k+2)=a(m+1) \end{gathered}$$

Given that, $a$ and $c$ are collinear vectors,

So, $k+2=0$ and $m+1=0$

$\Rightarrow k=-2$ and $\Rightarrow m=-1$

Replace $k=-2$ in $a+3b=kc$, we get

$$\begin{gathered} \Rightarrow a+3b=-2c \\ \Rightarrow a+3b+2c=0 \end{gathered}$$

Hence, the correct option is (A), i.e. $0$.