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If $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{c}=r \hat{i}+\hat{j}+(2 r-1 \hat{k}$ are three vectors such that $\vec{c}$ is parallel to the plane of $\vec{a}$ and $\vec{b}$, then $r$ is equal to
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Let $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}$, and
$$
\vec{c}=r \hat{i}+\hat{j}+(2 r-1) \hat{k}
$$
Since, $\vec{c}$ is parallel to the plane of $\vec{a}$ and $\vec{b}$ therefore, $\vec{a}, \vec{b}$ and $\vec{c}$ are coplanar.
$$
\begin{aligned}
& \therefore\left|\begin{array}{ccc}
1 & -2 & 3 \\
2 & 3 & -1 \\
r & 1 & 2 r-1
\end{array}\right|=0 \\
& \Rightarrow 1(6 r-3+1)+2(4 r-2+r)+ \\
& \Rightarrow 6 r-2+10 r-4+6-9 r=0 \\
& \Rightarrow r=0
\end{aligned}
$$
$$
\vec{c}=r \hat{i}+\hat{j}+(2 r-1) \hat{k}
$$
Since, $\vec{c}$ is parallel to the plane of $\vec{a}$ and $\vec{b}$ therefore, $\vec{a}, \vec{b}$ and $\vec{c}$ are coplanar.
$$
\begin{aligned}
& \therefore\left|\begin{array}{ccc}
1 & -2 & 3 \\
2 & 3 & -1 \\
r & 1 & 2 r-1
\end{array}\right|=0 \\
& \Rightarrow 1(6 r-3+1)+2(4 r-2+r)+ \\
& \Rightarrow 6 r-2+10 r-4+6-9 r=0 \\
& \Rightarrow r=0
\end{aligned}
$$
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