Three non-zero non-collinear vectors $\hat{\mathbf{a}}, \mathbf{b}$ and $\hat{\mathbf{c}}$ are such that…
Three non-zero non-collinear vectors $\hat{\mathbf{a}}, \mathbf{b}$ and $\hat{\mathbf{c}}$ are such that $\hat{\mathbf{a}}+3 \hat{\mathbf{b}}$ is collinear with $\hat{\mathbf{c}}$, while $\hat{\mathbf{c}}$ is $3 \hat{\mathbf{b}}+2 \hat{\mathbf{c}}$ collinear with $\hat{\mathbf{a}}$. Then $\hat{\mathbf{a}}+3 \hat{\mathbf{b}}+2 \hat{\mathbf{c}}$ equals to
$0$
$2 \hat{\mathbf{a}}$
$3 \hat{\mathbf{b}}$
$4 \hat{\mathbf{c}}$
Solution
Given, $\mathbf{a}+3 \mathbf{b}$ is collinear with c.
$
\begin{array}{rrr}
\therefore & a+3 b=\lambda c \\
\text { or } & a+3 b-\lambda c=0
\end{array}
$
And $3 \mathbf{b}+2 \mathbf{c}$ is collinear with $\mathbf{a}$.
$
\begin{aligned}
\therefore \quad 3 b+2 c & =\mu \mathrm{a} \\
3 b+2 c-\mu a & =0
\end{aligned}
$
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
$
\mathbf{a}+3 \mathbf{b}+2 \mathbf{c}=0
$