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If $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ are coplanar, then $\lambda$ is equal to :
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The correct answer is:
$-3 / 2$
Since $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}, 3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ are coplanar, then
$\left|\begin{array}{ccc}1 & -2 & 0 \\ 0 & 3 & 1 \\ \lambda & 3 & 0\end{array}\right|=0$
$\Rightarrow \quad 1\left|\begin{array}{ll}3 & 1 \\ 3 & 0\end{array}\right|+2\left|\begin{array}{ll}0 & 1 \\ \lambda & 0\end{array}\right|=0$
$\Rightarrow \quad-3+2(-\lambda)=0$
$\Rightarrow \quad \lambda=-\frac{3}{2}$
$\left|\begin{array}{ccc}1 & -2 & 0 \\ 0 & 3 & 1 \\ \lambda & 3 & 0\end{array}\right|=0$
$\Rightarrow \quad 1\left|\begin{array}{ll}3 & 1 \\ 3 & 0\end{array}\right|+2\left|\begin{array}{ll}0 & 1 \\ \lambda & 0\end{array}\right|=0$
$\Rightarrow \quad-3+2(-\lambda)=0$
$\Rightarrow \quad \lambda=-\frac{3}{2}$
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