A) \[-4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\]
B) \[4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\]
C) \[4\mathbf{i}-5\mathbf{j}+11\mathbf{k}\]
D) \[4\mathbf{i}-5\mathbf{j}-11\mathbf{k}\]
Correct Answer: B
Solution :
The line of intersection of the planes \[\mathbf{r}.(\mathbf{i}-3\mathbf{j}+\mathbf{k})=1\] and \[\mathbf{r}.(2\mathbf{i}+5\mathbf{j}-3\mathbf{k})=2\] is perpendicular to each of the normal vectors \[{{\mathbf{n}}_{1}}=\mathbf{i}-3\mathbf{j}+\mathbf{k}\] and \[{{\mathbf{n}}_{2}}=2\mathbf{i}+5\mathbf{j}-3\mathbf{k}\] \[\therefore \] It is parallel to the vector \[{{\mathbf{n}}_{1}}\times {{\mathbf{n}}_{2}}=(\mathbf{i}-3\mathbf{j}+\mathbf{k})\times (2\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] \[=\left. \left| \begin{matrix} \,\mathbf{i}\, \\ 1 \\ 2 \\ \end{matrix}\begin{matrix} \,\,\,\,\,\mathbf{j} \\ \,\,-3 \\ \,\,\,\,\,\,5 \\ \end{matrix}\begin{matrix} \,\,\,\,\mathbf{k} \\ \,\,\,1 \\ \,\,-3\, \\ \end{matrix} \right. \right|\]= \[4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\].
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