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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
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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
$$
\mathbf{a}+3 \mathbf{b}+2 \mathbf{c}=0
$$
$$
\begin{array}{rrr}
\therefore & a+3 b=\lambda c \\
\text { or } & a+3 b-\lambda c=0
\end{array}
$$
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
$$
\mathbf{a}+3 \mathbf{b}+2 \mathbf{c}=0
$$
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