$\vec{a}=\hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+\hat{j}-2 \hat{k}, \vec{c}=2 \hat{i}-3 \hat{j}-\hat{k},…

$\vec{a}=\hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+\hat{j}-2 \hat{k}, \vec{c}=2 \hat{i}-3 \hat{j}-\hat{k}, \vec{d}=2 \hat{i}+\hat{j}+\hat{k}$ are four vectors then $(\vec{a} \times \vec{c}) \times(\vec{b} \times \vec{d})=$
  1. $2 \hat{i}+19 \hat{j}-11 \hat{k}$
  2. $-8 \hat{i}+19 \hat{j}-29 \hat{k}$
  3. $2 \hat{i}+\hat{j}-11 \hat{k}$
  4. $-8 \hat{i}+\hat{j}-29 \hat{k}$

Solution

$\begin{aligned} & \text { } \vec{a} \times \vec{c}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 1 \\ 2 & -3 & -1\end{array}\right|=4 \hat{i}+3 \hat{j}-\hat{k} \\ & \vec{b} \times \vec{d}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -2 \\ 2 & 1 & 1\end{array}\right|=3 \hat{i}-5 \hat{j}-\hat{k} \\ & (\vec{a} \times \vec{c}) \times(\vec{b} \times \vec{d})=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 4 & 3 & -1 \\ 3 & -5 & -1\end{array}\right|=-8 \hat{i}+\hat{j}-29 \hat{k}\end{aligned}$

Asked in: AP EAMCET 2024 (21 May Shift 1)

Practice more Vectors questions on Aicharya