$\left|\begin{array}{lll}{1} & {2} & {4} \\ {1} & {\lambda} & {4} \\ {2} & {4} & {\lambda^{2}-1}\end{array}\right|_{R_{3} \rightarrow R_{3}-2 R_{1}}=0$
$\left|\begin{array}{ccc}{1} & {2} & {4} \\ {1} & {\lambda} & {4} \\ {0} & {0} & {\lambda^{2}-9}\end{array}\right|=0$
$\left(\lambda^{2}-9\right)(\lambda-2)=0$
$\Rightarrow \lambda=2 \quad$ OR $\quad \lambda^{2}=5$
$\vec{a} \times \vec{c}=\left|\begin{array}{lll}{\hat{i}} & {\hat{j}} & {\hat{k}} \\ {1} & {2} & {4} \\ {2} & {4} & {\lambda^{2}-1}\end{array}\right|$
$=\hat{i}\left(2 \lambda^{2}-2-16\right)-\hat{j}\left(\lambda^{2}-1-8\right)$
$\left.=\left(\lambda^{2}-9\right)(2 \hat{i}-\hat{j})=-5(2 \hat{i}-\hat{j}) \quad \text { (put } \lambda=2\right)$
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