Question

Let $a,b,c$ be three non-zero vectors such that no two of these are collinear. If the vectors $a+2b$ is collinear with $c$ and $b+3c$ is collinear with $a$ ($\lambda$ being some non-zero scalar), then $a+2b+6c$ equals to

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

Answer 1

Answer: (D)

$0$

Since, $a+2b$ is collinear with $c$
$a+2b=mc...\left(i\right)$
Since $b+3c$ is collinear with $a$
$b+3c=na.....\left(ii\right)$
Myltiplying Eq(ii) by $2$ and subtracting Eq.(ii) from Eq.(i), we get
$a-bc=mc-2na$
On comparing, we get
$m=-6;n=-\frac{1}{2}$
From Eq.(i), we have
$a+2b=-6c$
$\Rightarrow a+b+6c=0$