Question

Determine the value of λ for which the following planes are perpendicular to each ot

 2x − 4y + 3z = 5 and x + 2y + λz = 5

Answer 1

` \text{ We know that the planes } a_1 x + b_1 y + c_1 z + d_1 = 0 and a_2 x + b_2 y + c_2 z + d_2 = 0 \text{ are perperndicular to each other only if} `
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\]
\[\text{ The given planes are } 2x - 4y + 3z = 5 \text{ and }  x + 2y + \lambda z = 5 . \]
\[ \Rightarrow a_1 = 2; b_1 = - 4; c_1 = 3; a_2 = 1; b_2 = 2; c_2 = \lambda\]
\[\text{ It is given that the given planes are perpendicular } .\]
\[ \Rightarrow a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\]
\[ \Rightarrow \left( 2 \right) \left( 1 \right) + \left( - 4 \right) \left( 2 \right) + \left( 3 \right) \left( \lambda \right) = 0\]
\[ \Rightarrow 2 - 8 + 3\lambda = 0\]
\[ \Rightarrow 3\lambda = 6\]
\[ \Rightarrow \lambda = 2\]