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

$\text{The 2 planes are said to be perpendicular if the dot product of their normals is 0, they are}\\\text{said to be parallel if for planes $a_1x+a_2y+a_3z=c$ and $b_1x+b_2y+b_3z=d$ the }\\\text{ratios}\\\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}. \\\text{Using the above we check if the planes in a, b or c are perpendicular or parallel}\\a. \text{The normals here are $n_1 = (1,1,3)$ and $n_2 = (1,2,-1)$}. \text{ The dot product of $n_1$ and }\\\text{$n_2$ is given by }\\1(1) +1(2)+3(-1)=0\\\text{Since the dot product between $n_1$ and $n_2$ is 0, the planes are perpendicular.}\\b.\text{We notice that the dot product of the normals (3,1,1) and (-1,2,1) is not 0}\\\text{Also the ratios of the normals are $\frac{3}{4}$ and $\frac{-2}{2}$ and $\frac{1}{-4}$ are not equal.}\\\text{Therefore the plane is neither perpendicular or parallel.}\\c.\text{The dot product of the normals (3,1,1) and (-1,2,1) is 3(-1)+1(2)+1(1)=0}\\\text{Therefore the plane is perpendicular.}$