Question

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

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

Answer 1

Answer: (D)

If $a+2b$ is collinear with $c$, then
$a+2b=tc.....$(i)
Also, if $b+3c$ is collinear with $a$, then
$b+3c=\lambda a$
$\Rightarrow b=\lambda a-3c....$(ii)
On putting the value of $b$ in Eq. (i) we get
$a+2(\lambda a-3c)=tc$
$\Rightarrow (a-6c)=tc-2\lambda a$
On comparing, we get $1=-2\lambda$ and $-6=t$
$\Rightarrow \lambda =-\frac{1}{2}$
and $t=-6$
From Eq. (i) $a+2b=-6c$
$\Rightarrow a+2b+6c=0$