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Question
Find the equation of the plane passing through (a, b, c) and parallel to the plane \[\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .\]
Solution
\[ \text{ Let the equation of a plane parallel to the given plane be } \]
\[ \vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = k \]
\[\left( x \hat{i} + y \hat{j}+ z \hat{k} \right) . \left( \hat{i} + \hat{j} + \hat{k} \right) = k . . . \left( 1 \right)\]
\[ \text{ This passes through (a, b, c)} .\hspace{0.167em} \text{ So } ,\]
\[\left( a \hat{i} + b \hat{j} + c \hat{k} \right) . \left( \hat{i} + \hat{j} + \hat{k} \right) = k\]
\[ \Rightarrow k = a + b + c\]
\[ \text{ Substituting this in (1), we get } \]
\[\left( x \hat{i} + y \hat{j} + z \hat{k} \right) . \left( \hat{i} + \hat{j} + \hat{k} \right) = a + b + c\]
\[x + y + z = a + b + c, \text{ which is the equation of the required plane } .\]
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