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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2000

  • question_answer
    For any vector\[\vec{r}\],the value of\[\hat{i}\times (\overrightarrow{r}\times \hat{i})+\hat{j}\times (\overrightarrow{r}\times \hat{j})+\hat{k}\times (\overrightarrow{r}\times \hat{k})\]is:

    A)  \[\overrightarrow{0}\]                                

    B)  \[2\overrightarrow{r}\]

    C)  \[-2\overrightarrow{r}\]                             

    D)         \[3\overrightarrow{r}\]

    E)  none of these

    Correct Answer: B

    Solution :

    Let\[\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}\] \[\therefore \]\[\hat{i}\times (\overrightarrow{r}\times \hat{i})+\hat{j}\times (\overrightarrow{r}\times \hat{j})+\hat{k}\times (\hat{r}\times \hat{k})\] \[=(\hat{i}.\hat{i})\overrightarrow{r}-(\hat{i}.\overrightarrow{r})\hat{i}+(\hat{j}.\hat{j})\overrightarrow{r}-(\hat{j}.\overrightarrow{r})\hat{j}\]                                                 \[+(\hat{k}.\hat{k})\overrightarrow{r}-(\hat{k}-\overrightarrow{r})\hat{k}\] \[=3\overrightarrow{r}-(x\hat{i}+y\hat{j}+z\hat{k})\] \[=3\overrightarrow{r}-\overrightarrow{r}\] \[=2\overrightarrow{r}\]


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