1.x+y+3z+10=0 and x+2y-z=1
n1=⟨1,1,3⟩,n2=⟨1,2,−1⟩11=12 The vectors n1 and n2 are not collinear.
n1⋅n2=1(1)+1(2)+3(−1)=0The vectors n1 and n2 are orthogonal.
Hence, the given planes are perpendicular.
2.3x-2y+z-6=0 and 4x+2y-4z=0
n1=⟨3,−2,1⟩,n2=⟨4,2,−4⟩34=−22 The vectors n1 and n2 are not collinear.
n1⋅n2=3(4)+(−2)(2)+1(−4)=4=0The vectors n1 and n2 are not orthogonal.
Hence, the given planes are neither parallel nor perpendicular.
3.3x+y+z-1=0 and -x+2y+z+3=0
n1=⟨3,1,1⟩,n2=⟨−1,2,1⟩3−1=12 The vectors n1 and n2 are not collinear.
n1⋅n2=3(−1)+1(2)+1(1)=0The vectors n1 and n2 are orthogonal.
Hence, the given planes are perpendicular.
4.x-3y+z+1=0 and 3x-4y+z-1=0
n1=⟨1,−3,1⟩,n2=⟨3,−4,1⟩13=−3−4 The vectors n1 and n2 are not collinear.
n1⋅n2=1(3)+(−3)(−4)+1(1)=16=0The vectors n1 and n2 are not orthogonal.
Hence, the given planes are neither parallel nor perpendicular.
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