If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=2…

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, then $[(\mathbf{a} \times \mathbf{b}) \times(\mathbf{b} \times \mathbf{c})(\mathbf{b} \times \mathbf{c}) \times(\mathbf{c} \times \mathbf{a})(\mathbf{c} \times \mathbf{a}) \times(\mathbf{a} \times \mathbf{b})]=$
  1. 160000
  2. -8000
  3. 400
  4. -40

Solution

We have, $ \begin{aligned} & \mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}} \\ & \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}} \\ & \mathbf{c}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}} \end{aligned} $ Now, $ \begin{aligned} & (\mathbf{a} \times \mathbf{b}) \times(\mathbf{b} \times \mathbf{c})=[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{b}-[\mathbf{a b} \mathbf{b}] \mathbf{c}=[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{b} \\ & (\mathbf{b} \times \mathbf{c}) \times(\mathbf{c} \times \mathbf{a})=[\mathbf{b} \mathbf{c} \mathbf{a}] \mathbf{c}-[\mathbf{b} \mathbf{c}] \mathbf{a}=[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{c} \\ & (\mathbf{c} \times \mathbf{a}) \times(\mathbf{a} \times \mathbf{b})=[\mathbf{c} \mathbf{a} \mathbf{b}] \mathbf{a}-[\mathbf{c} \mathbf{a} \mathbf{a}] \mathbf{b}=[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{a} \\ & \text { Now, } \quad[\mathbf{a} \mathbf{b} \mathbf{c}]=\left[\begin{array}{ccc} 1 & -2 & -3 \\ 2 & 1 & -1 \\ 1 & 3 & -2 \end{array} \mid\right. \\ & =1(-2+3)+2(-4+1)-3(6-1)=1-6-15=-20 \end{aligned} $ $\begin{aligned} & \therefore[(\mathbf{a} \times \mathbf{b}) \times(\mathbf{b} \times \mathbf{c})(\mathbf{b} \times \mathbf{c}) \times(\mathbf{c} \times \mathbf{a}) \\ & =\left[\begin{array}{ll}-20 \mathbf{b}-20 \mathbf{c}-20 \mathbf{a}] \\ (\mathbf{c} \times \mathbf{a}) \times(\mathbf{a} \times \mathbf{b})]\end{array}\right. \\ & =\left|\begin{array}{ccc}-40 & -20 & 20 \\ -20 & -60 & 40 \\ -20 & 40 & 60\end{array}\right| \\ & =-40[-3600-1600]+20[-1200+800] \\ & +20[-800-1200] \\ & =208000-8000-40000=160000 .\end{aligned}$

Asked in: AP EAMCET 2018 (22 Apr Shift 1)

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