A) \[5(\hat{j}-\hat{k})\]
B) \[\hat{i}+7\hat{j}-\hat{k}\]
C) \[5(\hat{j}+\hat{k})\]
D) \[2\hat{i}-7\hat{j}-\hat{k}\]
E) \[5(\hat{i}+\hat{k})\]
Correct Answer: A
Solution :
Any vector\[\bot \]to\[\overrightarrow{a}\]and coplanar to\[\overrightarrow{b}\]and\[\overrightarrow{c}\]is given by\[\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})\] \[\therefore \]Required vector is \[(2\hat{i}+\hat{j}+\hat{k})\times [(\hat{i}+2\hat{j}+\hat{k})\times (\hat{i}+\hat{j}+2\hat{k})]\] \[=(2\hat{i}+\hat{j}+\hat{k})\times \left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{matrix} \right|\] \[=(2\hat{i}+\hat{j}+\hat{k})\times (3\hat{i}-\hat{j}-\hat{k})\] \[=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 2 & 1 & 1 \\ 3 & -1 & -1 \\ \end{matrix} \right|=5(\hat{j}-\hat{k})\]
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