MHT CET · Maths · Vector Algebra
If \(\bar{a}=2 i+3 j-\hat{k}, \bar{b}=-i+2 j-4 \hat{k}\) and \(\bar{c}=i+j+\hat{k}\), then \((\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=\)
- A \(-74\)
- B 64
- C \(-64\)
- D 74
Answer & Solution
Correct Answer
(A) \(-74\)
Step-by-step Solution
Detailed explanation
\(\bar{a} \times \bar{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ -1 & 2 & -4\end{array}\right|=\hat{i}(-12+2)-\hat{j}(-8-1)~+\) \(\hat{k}(4+3)=-10 \hat{i}+9 \hat{j}+7 \hat{k}\)
\(\bar{a} \times \bar{c}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1\end{array}\right|=\hat{i}(3+1)-\hat{j}(2+1)~+\) \(\hat{k}(2-3)=4 \hat{i}-3 \hat{j}-\hat{k}\)
\(\text{Here}\) \(\quad(\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=(-10 \hat{i}+9 \hat{j}+7 \hat{k}) \cdot(4 \hat{i}-3 \hat{j}-\hat{k})\)
\(=(-10)(4)+9(-3)+7(-1)\)
\(=-40-27-7=-74\)
\(\bar{a} \times \bar{c}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1\end{array}\right|=\hat{i}(3+1)-\hat{j}(2+1)~+\) \(\hat{k}(2-3)=4 \hat{i}-3 \hat{j}-\hat{k}\)
\(\text{Here}\) \(\quad(\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=(-10 \hat{i}+9 \hat{j}+7 \hat{k}) \cdot(4 \hat{i}-3 \hat{j}-\hat{k})\)
\(=(-10)(4)+9(-3)+7(-1)\)
\(=-40-27-7=-74\)
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