A) \[\overrightarrow{r}=(-\hat{i}+\hat{j}-\hat{k})+t(-\hat{i}-2\hat{j}+4\hat{k})\]
B) \[\overrightarrow{r}=(-\hat{i}+\hat{j}-\hat{k})+t(\hat{i}-2\hat{j}+4\hat{k})\]
C) \[\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+t(-\hat{i}-4\hat{j}+7\hat{k})\]
D) \[\overrightarrow{r}=(-\hat{i}+\hat{j}-\hat{k})+t(-2\hat{i}+2\hat{j}+4\hat{k})\]
E) \[\overrightarrow{r}=(-\hat{i}+\hat{j}-3\hat{k})+t(2\hat{i}+6\hat{j}-8\hat{k})\]
Correct Answer: A
Solution :
Given plane is \[\Rightarrow \]\[\overrightarrow{r}=r(\hat{i}+2\hat{j}-4\hat{k})+t(3\hat{i}+4\hat{j}-4\hat{k})\] \[+(1-t)(2\hat{i}-7\hat{j}-3\hat{k})\] \[\Rightarrow \]\[\overrightarrow{r}=(2\hat{i}-7\hat{j}+3\hat{k})+s(\hat{i}+2\hat{j}-4\hat{k})\] \[+t(\hat{i}+11\hat{j}-\hat{k})\] Comparing it with the equation of plane \[\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}+\mu \overrightarrow{c},\]we get \[\overrightarrow{b}=\hat{i}+2\hat{j}-4\hat{k}\] and \[\overrightarrow{c}=\hat{i}+11\hat{j}-\hat{k}\] Now, \[\overrightarrow{b}\times \overrightarrow{c}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 2 & -4 \\ 1 & 11 & -1 \\ \end{matrix} \right|\] \[=42\hat{i}-3\hat{j}+9\hat{k}\] \[\therefore \]Parametric form of plane is \[\overrightarrow{r}.(\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})\] \[\Rightarrow \] \[\overrightarrow{r}.(42\hat{i}-3\hat{j}+9\hat{k})\] \[=(2\hat{i}-7\hat{j}+3\hat{k}).(42\hat{i}-3\hat{j}+9\hat{k})\] which is of the form \[\overrightarrow{r}.\overrightarrow{r}=d\] \[\Rightarrow \] \[\overrightarrow{r}=42\hat{i}-3\hat{j}+9\hat{k}\] Now, the line given in option (a) is \[\overrightarrow{r}=(-\hat{i}+\hat{j}+\hat{k})+t(-\hat{i}-2\hat{j}+4\hat{k})\] Comparing it with\[\overrightarrow{r}=\overrightarrow{p}+t\overrightarrow{q},\]we get \[\overrightarrow{q}=(-\hat{i}-2\hat{j}+4\hat{k})\] Since, \[\overrightarrow{q}.\overrightarrow{n}=(-\hat{i}-2\hat{j}+4\hat{k}).(42\hat{i}-3\hat{j}+9\hat{k})\] \[=-42+6+36=0\] Hence, the line given in option (a) is parallel to the given plane.
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