1.x+y+3z+10=0 and x+2y-z=1
n ⃗ 1 = ⟨ 1 , 1 , 3 ⟩ , n ⃗ 2 = ⟨ 1 , 2 , − 1 ⟩ \vec n_1=\langle1,1,3\rangle, \vec n_2=\langle1,2,-1\rangle n 1 = ⟨ 1 , 1 , 3 ⟩ , n 2 = ⟨ 1 , 2 , − 1 ⟩ 1 1 ≠ 2 1 \dfrac{1}{1}\not=\dfrac{2}{1} 1 1 = 1 2 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are not collinear.
n ⃗ 1 ⋅ n ⃗ 2 = 1 ( 1 ) + 1 ( 2 ) + 3 ( − 1 ) = 0 \vec n_1\cdot\vec n _2=1(1)+1(2)+3(-1)=0 n 1 ⋅ n 2 = 1 ( 1 ) + 1 ( 2 ) + 3 ( − 1 ) = 0 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are orthogonal.
Hence, the given planes are perpendicular.
2.3x-2y+z-6=0 and 4x+2y-4z=0
n ⃗ 1 = ⟨ 3 , − 2 , 1 ⟩ , n ⃗ 2 = ⟨ 4 , 2 , − 4 ⟩ \vec n_1=\langle3,-2,1\rangle, \vec n_2=\langle4,2,-4\rangle n 1 = ⟨ 3 , − 2 , 1 ⟩ , n 2 = ⟨ 4 , 2 , − 4 ⟩ 4 3 ≠ 2 − 2 \dfrac{4}{3}\not=\dfrac{2}{-2} 3 4 = − 2 2 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are not collinear.
n ⃗ 1 ⋅ n ⃗ 2 = 3 ( 4 ) + ( − 2 ) ( 2 ) + 1 ( − 4 ) = 4 ≠ 0 \vec n_1\cdot\vec n _2=3(4)+(-2)(2)+1(-4)=4\not=0 n 1 ⋅ n 2 = 3 ( 4 ) + ( − 2 ) ( 2 ) + 1 ( − 4 ) = 4 = 0 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are not orthogonal.
Hence, the given planes are neither parallel nor perpendicular.
3.3x+y+z-1=0 and -x+2y+z+3=0
n ⃗ 1 = ⟨ 3 , 1 , 1 ⟩ , n ⃗ 2 = ⟨ − 1 , 2 , 1 ⟩ \vec n_1=\langle3,1,1\rangle, \vec n_2=\langle-1,2,1\rangle n 1 = ⟨ 3 , 1 , 1 ⟩ , n 2 = ⟨ − 1 , 2 , 1 ⟩ − 1 3 ≠ 2 1 \dfrac{-1}{3}\not=\dfrac{2}{1} 3 − 1 = 1 2 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are not collinear.
n ⃗ 1 ⋅ n ⃗ 2 = 3 ( − 1 ) + 1 ( 2 ) + 1 ( 1 ) = 0 \vec n_1\cdot\vec n _2=3(-1)+1(2)+1(1)=0 n 1 ⋅ n 2 = 3 ( − 1 ) + 1 ( 2 ) + 1 ( 1 ) = 0 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are orthogonal.
Hence, the given planes are perpendicular.
4.x-3y+z+1=0 and 3x-4y+z-1=0
n ⃗ 1 = ⟨ 1 , − 3 , 1 ⟩ , n ⃗ 2 = ⟨ 3 , − 4 , 1 ⟩ \vec n_1=\langle1,-3,1\rangle, \vec n_2=\langle3,-4,1\rangle n 1 = ⟨ 1 , − 3 , 1 ⟩ , n 2 = ⟨ 3 , − 4 , 1 ⟩ 3 1 ≠ − 4 − 3 \dfrac{3}{1}\not=\dfrac{-4}{-3} 1 3 = − 3 − 4 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are not collinear.
n ⃗ 1 ⋅ n ⃗ 2 = 1 ( 3 ) + ( − 3 ) ( − 4 ) + 1 ( 1 ) = 16 ≠ 0 \vec n_1\cdot\vec n _2=1(3)+(-3)(-4)+1(1)=16\not=0 n 1 ⋅ n 2 = 1 ( 3 ) + ( − 3 ) ( − 4 ) + 1 ( 1 ) = 16 = 0 The vectors n ⃗ 1 \vec n_1 n 1 and n ⃗ 2 \vec n_2 n 2 are not orthogonal.
Hence, the given planes are neither parallel nor perpendicular.
Comments