If \(\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \quad \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\), \(\overline{\mathrm{c}}=\mathrm{c}_1 \hat{i}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) and \(\left[\begin{array}{lll}3 \overline{\mathrm{a}}+\overline{\mathrm{b}} & 3 \overline{\mathrm{~b}}+\overline{\mathrm{c}} & 3 \overline{\mathrm{c}}+\overline{\mathrm{a}}\end{array}\right]\) \(=\lambda\left|\overline{\mathrm{a}} \cdot \hat{\mathrm{i}} \overline{\mathrm{a}} \cdot \hat{\mathrm{j}} \overline{\mathrm{a}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{b}} \cdot \hat{\mathrm{i}} \overline{\mathrm{b}} \cdot \hat{\mathrm{j}} \overline{\mathrm{b}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{c}} \cdot \hat{\mathrm{i}} \overline{\mathrm{c}} \cdot \hat{\mathrm{j}} \overline{\mathrm{c}} \cdot \hat{\mathrm{k}}\right|\) then the value of \(\lambda\) is

\(\lambda\left|\begin{array}{lll}\overline{\mathrm{a}} \cdot \hat{\mathrm{i}} & \overline{\mathrm{a}} \cdot \hat{\mathrm{j}} & \overline{\mathrm{a}} \cdot \hat{\mathrm{k}} \\ \overline{\mathrm{b}} \cdot \hat{\mathrm{i}} & \overline{\mathrm{b}} \cdot \hat{\mathrm{j}} & \overline{\mathrm{b}} \cdot \hat{\mathrm{k}} \\ \overline{\mathrm{c}} \cdot \hat{\mathrm{i}} & \overline{\mathrm{c}} \cdot \hat{\mathrm{j}} & \overline{\mathrm{c}} \cdot \hat{\mathrm{k}}\end{array}\right|=\lambda\left|\begin{array}{lll}\mathrm{a}_1 & \mathrm{a}_2 & \mathrm{a}_3 \\ \mathrm{~b}_1 & \mathrm{~b}_2 & \mathrm{~b}_3 \\ \mathrm{c}_1 & \mathrm{c}_2 & \mathrm{c}_3\end{array}\right|=\lambda\left[\begin{array}{lll}\mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right]\)
\(\left[\begin{array}{lll}3 \overline{\mathrm{a}}+\overline{\mathrm{b}} & 3 \overline{\mathrm{~b}}+\overline{\mathrm{c}} & 3 \overline{\mathrm{c}}+\overline{\mathrm{a}}\end{array}\right]\)
\(\begin{aligned} & =(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot((3 \overline{\mathrm{~b}}+\overline{\mathrm{c}}) \times(3 \overline{\mathrm{c}}+\overline{\mathrm{a}})) \\ & =(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot(9(\overline{\mathrm{~b}} \times \overline{\mathrm{c}})+3(\overline{\mathrm{~b}} \times \overline{\mathrm{a}})+3(\overline{\mathrm{c}} \times \overline{\mathrm{c}})+(\overline{\mathrm{c}} \times \overline{\mathrm{a}}))\end{aligned}\)
\(\begin{aligned} & =(3 \overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot(9(\overline{\mathrm{~b}} \times \overline{\mathrm{c}})+3(\overline{\mathrm{~b}} \times \overline{\mathrm{a}})+0+(\overline{\mathrm{c}} \times \overline{\mathrm{a}})) \\ & =27 \overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})+0+0+0+0+\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}}) \\ & =27\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]+\left[\begin{array}{lll}\overline{\mathrm{b}} & \overline{\mathrm{c}} & \overline{\mathrm{a}}\end{array}\right] \\ & =27\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]+\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right] \\ & =28\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]\end{aligned}\)
\(\therefore \quad \lambda=28\)