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If the vectors $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathrm{c}=x \hat{\mathbf{i}}+(x-2) \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are coplanar, then $x=$
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Verified Answer
The correct answer is:
-2
Given, $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$,
$\mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$
and
$\mathbf{c}=x \hat{\mathbf{i}}+(x-2) \hat{\mathbf{j}}-\hat{\mathbf{k}}$
Since, the given vectors are coplanar, therefore
$$
\Rightarrow\left|\begin{array}{ccc}
{\left[\begin{array}{ccc}
\mathbf{a} & \mathbf{b} & \mathbf{c}
\end{array}\right]=0} \\
1 & 1 & 1 \\
1 & -1 & 2 \\
x & x-2 & -1
\end{array}\right|=0
$$
$$
\Rightarrow 5-2 x+1+2 x+2 x-2=0
$$
$$
\begin{aligned}
\Rightarrow & 2 x+4 & =0 \\
\Rightarrow & 2 x & =-4 \\
\Rightarrow & x & =-2
\end{aligned}
$$
$\mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$
and
$\mathbf{c}=x \hat{\mathbf{i}}+(x-2) \hat{\mathbf{j}}-\hat{\mathbf{k}}$
Since, the given vectors are coplanar, therefore
$$
\Rightarrow\left|\begin{array}{ccc}
{\left[\begin{array}{ccc}
\mathbf{a} & \mathbf{b} & \mathbf{c}
\end{array}\right]=0} \\
1 & 1 & 1 \\
1 & -1 & 2 \\
x & x-2 & -1
\end{array}\right|=0
$$
$$
\Rightarrow 5-2 x+1+2 x+2 x-2=0
$$
$$
\begin{aligned}
\Rightarrow & 2 x+4 & =0 \\
\Rightarrow & 2 x & =-4 \\
\Rightarrow & x & =-2
\end{aligned}
$$
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