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Question
A vector parallel to the line of intersection of the planes\[\vec{r} \cdot \left( 3 \hat{i} - \hat{j} + \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( \hat{i} + 4 \hat{j} - 2 \hat{k} \right) = 2\] is
Options
\[- 2 \hat{i} + 7 \hat{j}+ 13 \hat{k} \]
\[2 \hat{i} + 7 \hat{j} - 13 \hat{k}\]
\[-2 \hat{i} + 7 \hat{j} + 13 \hat{k}\]
\[2 \hat{i} + 7 \hat{j} + 13 \hat{k}\]
Solution
\[2 \hat{i} + 7 \hat{j} - 13 \hat{k}\]
\[\text{ Let the required vector be a } \hat { i } + b \hat{j} + c \hat{ k } . . . \left( 1 \right)\]
\[\text{ Since the vector is parallel to the line of intersection of the given planes } ,\]
\[3a - b + c = 0 . . . \left( 2 \right)\]
\[a + 4b - 2c = 0 . . . \left( 3 \right)\]
\[\text{ Solving (2) and (3), we get} \]
\[\frac{a}{- 2} = \frac{b}{7} = \frac{c}{13}\]
\[\text{ Substituting these values in (1), we get } \]
\[ - 2 \hat{i} + 7 \hat{j} + 13 \hat{k} , \text{ which is the required vector } .\]
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