If \(\vec{a}=\lambda \hat{i}+\hat{j}-2\hat{k}\) , \(\vec{b}=\hat{i}+\lambda \hat{j}-2\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\) and \(\begin{bmatrix}{\vec{a}} & {\vec{b}} & {\vec{c}} \end{bmatrix}=7\), then the values of the \(\lambda\) are
\[ \begin{aligned} & \text{Given that } [a \hat{b c} \hat{c} \uparrow = 7, \\ & \quad \left| \begin{array}{ccc} \lambda & 1 & -2 \\ 1 & \lambda & -2 \end{array} \right| = 7. \\ & \Rightarrow \lambda(\lambda + 2) - 1(1 + 2) - 2(1 - \lambda) = 7, \\ & \text{or } \lambda^2 + 4\lambda - 12 = 0. \\ & \Rightarrow (\lambda - 2)(\lambda + 6) = 0, \\ & \Rightarrow \lambda = 2, -6. \end{aligned} \]
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