question_answer

If \[\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{b}\]and \[\overrightarrow{r}.\overrightarrow{a}=0,\] where \[\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k},\] \[\overrightarrow{b}=3\hat{i}-\hat{j}+\hat{k},\] and \[\overrightarrow{c}=\hat{i}+\hat{j}+3\hat{k},\] then \[\overrightarrow{r}\] is equal to _______.

A)  \[\left( -\hat{i}+\hat{j}+\hat{k} \right)\]             

B)  \[2\left( -\hat{i}+\hat{j}+\hat{k} \right)\]

C)  \[\left( -\hat{i}+\hat{j}-\hat{k} \right)\]               

D)  \[2\left( -\hat{i}+\hat{j}-\hat{k} \right)\]

E)  None of these