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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
\[\text{ Substituting } \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} \text{ in the given equation of the plane, we get } \]
\[\left( x \hat{i} + y \hat{j} + z \hat{k} \right) . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2\]
\[ \Rightarrow x + y + z - 2 = 0 . . . (1)\]
\[\text{ The equation of a plane which is parallel to plane (1) is of the form } \]
\[x + y + z = k . . . \left( 2 \right)\]
\[ \text{ It is given that plane (2) is passing through the point ( a, b, c ). So } ,\]
\[a + b + c = k\]
\[ \text{ Substituting this value of k in (2), we get } \]
\[\text{ x + y + z = a + b + c, which is the required equation of the plane} .\]
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