Question #213704

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

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

b) 3x - 2y + z - 6 = 0 and 4x + 2y - 4z = 0

c) 3x + y + z - 1 = 0 and -x + 2y + z + 3 = 0


1
Expert's answer
2021-07-23T12:23:05-0400

a) n1=(1,1,3),n2=(1,2,1)\vec n_1=(1,1,3), \vec n_2=(1, 2, -1)


n1n2=1(1)+1(2)+3(1)=0\vec n_1\cdot\vec n_2=1(1)+1(2)+3(-1)=0

The given planes are perpendicular.


b) n1=(3,2,1),n2=(4,2,4)\vec n_1=(3,-2,1), \vec n_2=(4, 2, -4)


n1n2=3(4)2(2)+1(4)=40\vec n_1\cdot\vec n_2=3(4)-2(2)+1(-4)=4\not=0

4322\dfrac{4}{3}\not=\dfrac{2}{-2}

The given planes are neither perpendicular nor parallel.


c) n1=(3,1,1),n2=(1,2,1)\vec n_1=(3,1,1), \vec n_2=(-1, 2, 1)


n1n2=3(1)+1(2)+1(1)=0\vec n_1\cdot\vec n_2=3(-1)+1(2)+1(1)=0

The given planes are perpendicular.




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