Question

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

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

Answer 1

Answer: (D)

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