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Question
Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
Solution
\[ \text{ The equation of any plane passing through point (-1, 3, 2) is } \]
\[a \left( x + 1 \right) + b \left( y - 3 \right) + c \left( z - 2 \right) = 0 . . . \left( 1 \right)\]
\[ \text{ It is given that (1) is perpendicular to the plane x + 2y + 3z = 5 . So, }\]
\[a + 2b + 3c = 0 . . . \left( 2 \right)\]
\[ \text{ It is given that (1) is perpendicular to the plane 3x + 3y + z = 0 . So } ,\]
\[3a + 3b + c = 0 . . . \left( 3 \right)\]
\[ \text{ Solving (1), (2) and (3), we get } \]
\[\begin{vmatrix}x + 1 & y - 3 & z - 2 \\ 1 & 2 & 3 \\ 3 & 3 & 1\end{vmatrix} = 0\]
\[ \Rightarrow - 7 \left( x + 1 \right) + 8 \left( y - 3 \right) - 3 \left( z - 2 \right) = 0\]
\[ \Rightarrow 7x - 8y + 3z + 25 = 0\]
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