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If the vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are coplanar, then the value of $\lambda$ is
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The correct answer is:
6
Given vectors $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are coplanar
$\left|\begin{array}{ccc}2 & -3 & 4 \\ 2 & 1 & -1 \\ \lambda & -1 & 2\end{array}\right|=0$
$\Rightarrow \quad 2(2-1)+3(4+\lambda)+4(-2-\lambda)=0$
$\Rightarrow \quad 2+12+3 \lambda-8-4 \lambda=0$
$\Rightarrow \quad 6-\lambda=0$
$\Rightarrow \quad \lambda=6$
$\left|\begin{array}{ccc}2 & -3 & 4 \\ 2 & 1 & -1 \\ \lambda & -1 & 2\end{array}\right|=0$
$\Rightarrow \quad 2(2-1)+3(4+\lambda)+4(-2-\lambda)=0$
$\Rightarrow \quad 2+12+3 \lambda-8-4 \lambda=0$
$\Rightarrow \quad 6-\lambda=0$
$\Rightarrow \quad \lambda=6$
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