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If the vectors $2 \hat{i}-\hat{j}-\hat{k} ; \hat{i}+2 \hat{j}-3 \hat{k}$ and $3 \hat{i}+\lambda \hat{j}+5 \hat{k}$ are coplanar, then the value of $\lambda$ is
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Verified Answer
The correct answer is:
-8
For given coplanar vectors, we write
$$
\begin{aligned}
& \left|\begin{array}{ccc}
2 & -1 & -1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=\Rightarrow 2(10+3 \lambda)+(14)-(\lambda-6)=0 \\
& \Rightarrow 5 \lambda+40=0 \Rightarrow \lambda=-8
\end{aligned}
$$
$$
\begin{aligned}
& \left|\begin{array}{ccc}
2 & -1 & -1 \\
1 & 2 & -3 \\
3 & \lambda & 5
\end{array}\right|=\Rightarrow 2(10+3 \lambda)+(14)-(\lambda-6)=0 \\
& \Rightarrow 5 \lambda+40=0 \Rightarrow \lambda=-8
\end{aligned}
$$
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