Question

Three non-zero non-collinear vectors $\hat{ a }, \hat{ b }$ and $\hat{ c }$ are such that $\hat{ a }+3 \hat{ b }$ is collinear with $\hat{ c }$, while $\hat{ c }$ is $3 \hat{ b }+2 \hat{ c }$ collinear with â. Then $\hat{ a }+3 \hat{ b }+2 \hat{ c }$ equals to

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

Answer 1

Answer: (A)

Given, $a +3 b$ is collinear with $c$.
$\therefore a +3 b =\lambda c$
Or $a +3 b -\lambda c =0\,\,\,...(i)$
And $3 b +2 c$ is collinear with $a$.
$\therefore 3 b +2 c =\mu a$
$3 b +2 c -\mu a =0\,\,\,...(ii)$
From Eqs. (i) and (ii), we get
$a +3 b -\lambda c =3 b +2 c -\mu a$
On equating $c$, we get
$\lambda=-2$
On putting $\lambda=-2$ in Eq. (i), we get
$a+3 b+2 c=0$