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Question
Find the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane \[\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2\]
Solution
The required plane passes through \[a \hat{i} + b \hat{j} + c \hat{k} \] and is parallel to the plane \[\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2\] So, it is normal to the vector \[\hat{i} + \hat{j} + \hat{k} \] which is normal to the given plane.
Hence, the vector equation of the required plane is
\[ \Rightarrow \vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = \left( a \hat{i} + b \hat{j} + c \hat{k} \right) . \left( \hat{i} + \hat{j} + \hat{k} \right)\]
\[ \Rightarrow \vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = a + b + c\]
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