Question #209806

Determine in each case whether the given planes are parallel or perpendicular:

1.x+y+3z+10=0 and x+2y-z=1,

2.3x-2y+z-6=0 and 4x+2y-4z=0,

3.3x+y+z-1=0 and -x+2y+z+3=0,

4.x-3y+z+1=0 and 3x-4y+z-1=0.


1
Expert's answer
2021-06-27T18:40:34-0400

1.x+y+3z+10=0 and x+2y-z=1


n1=1,1,3,n2=1,2,1\vec n_1=\langle1,1,3\rangle, \vec n_2=\langle1,2,-1\rangle1121\dfrac{1}{1}\not=\dfrac{2}{1}

The vectors n1\vec n_1 and n2\vec n_2 are not collinear.



n1n2=1(1)+1(2)+3(1)=0\vec n_1\cdot\vec n _2=1(1)+1(2)+3(-1)=0

The vectors n1\vec n_1 and n2\vec n_2 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\vec n_1=\langle3,-2,1\rangle, \vec n_2=\langle4,2,-4\rangle4322\dfrac{4}{3}\not=\dfrac{2}{-2}

The vectors n1\vec n_1 and n2\vec n_2 are not collinear.



n1n2=3(4)+(2)(2)+1(4)=40\vec n_1\cdot\vec n _2=3(4)+(-2)(2)+1(-4)=4\not=0

The vectors n1\vec n_1 and n2\vec 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


n1=3,1,1,n2=1,2,1\vec n_1=\langle3,1,1\rangle, \vec n_2=\langle-1,2,1\rangle1321\dfrac{-1}{3}\not=\dfrac{2}{1}

The vectors n1\vec n_1 and n2\vec n_2 are not collinear.



n1n2=3(1)+1(2)+1(1)=0\vec n_1\cdot\vec n _2=3(-1)+1(2)+1(1)=0

The vectors n1\vec n_1 and n2\vec n_2 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\vec n_1=\langle1,-3,1\rangle, \vec n_2=\langle3,-4,1\rangle3143\dfrac{3}{1}\not=\dfrac{-4}{-3}

The vectors n1\vec n_1 and n2\vec n_2 are not collinear.



n1n2=1(3)+(3)(4)+1(1)=160\vec n_1\cdot\vec n _2=1(3)+(-3)(-4)+1(1)=16\not=0

The vectors n1\vec n_1 and n2\vec n_2 are not orthogonal.

Hence, the given planes are neither parallel nor perpendicular.



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