Question

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

Answer 1

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

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

The given planes are perpendicular.

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

$$\vec n_1\cdot\vec n_2=3(4)-2(2)+1(-4)=4\not=0$$ $$\dfrac{4}{3}\not=\dfrac{2}{-2}$$

The given planes are neither perpendicular nor parallel.

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

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

The given planes are perpendicular.