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

If $\mathbf{a}=2 \mathbf{i}+\mathbf{j}-3 \mathbf{k}, \mathbf{b}=\mathbf{i}-2 \mathbf{j}+\mathbf{k}$ $\mathbf{c}=-\mathbf{i}+\mathbf{j}-4 \mathbf{k}$ and $\mathbf{d}=\mathbf{i}+\mathbf{j}+\mathbf{k}$, then $|(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})|=$
  1. $5 \sqrt{114}$
  2. $5 \sqrt{94}$
  3. $5 \sqrt{124}$
  4. $5 \sqrt{104}$

Solution

Given that, $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ $ \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}} $ $ \mathbf{c}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-4 \hat{\mathbf{k}} \text { and } \mathbf{d}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}} $ Now, $\mathbf{a} \times \mathbf{b}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 2 & 1 & -3 \\ 1 & -2 & 1\end{array}\right|$ $ \begin{aligned} & =\mathbf{i}(1-6)-\hat{\mathbf{j}}(2+3)+\hat{\mathbf{k}}(-4-1) \\ \mathbf{a} \times \mathbf{b} & =-5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-5 \hat{\mathbf{k}} \end{aligned} $ Now, $ \begin{aligned} \mathbf{c} \times \mathbf{d} & =\left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -1 & 1 & -4 \\ 1 & 1 & 1 \end{array}\right| \\ & =\hat{\mathbf{i}}(1+4)-\hat{\mathbf{j}}(-1+4)+\hat{\mathbf{k}}(-1-1) \\ & =5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}} \end{aligned} $ Now, $ \begin{aligned} (\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=\left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ -5 & -5 & -5 \\ 5 & -3 & -2 \end{array}\right| \\ =\hat{\mathbf{i}}(10-15)-\hat{\mathbf{j}}(10+25)+\hat{\mathbf{k}}(+15+25) \\ =-5 \hat{\mathbf{i}}-35 \hat{\mathbf{j}}+40 \hat{\mathbf{k}}=5(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+8 \hat{\mathbf{k}}) \end{aligned} $ $ \begin{aligned} |(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})| & =5 \sqrt{(-1)^2+(-7)^2+(8)^2} \\ & =5 \sqrt{1+49+64}=5 \sqrt{114} \end{aligned} $

Asked in: AP EAMCET 2018 (23 Apr Shift 1)

Practice more Vectors questions on Aicharya