Question

Determine in each case whether the given planes are parallel or perpendicular (2) (2) (a) x + y + 3z +10 - 0 and x + 2y - Z-1. (b) 3x - 2y + 2 - 6 - 0 and 4x +2y - 42 -0. (c) 3x + y +2 -1 = 0 and -x +2y +2 +3 = 0, (+0 (d) x - 3y + 2 + 1 - 0 and 3x - 4y +2 -1 -0. (2)

Answer 1

a)

$x+y+3z+10=0 => n_1=(1,1,3)\\ x+2y-z-1=0 => n_2=(1,2,-1)\\ (n_1,n_2)=1+2-3=0=> \cos(n_1,n_2)=0=>\\$ planes are perpendicular

b)

$3x-2y+2z-6=0=>n_1=(3,-2,2)\\ 4x+2y-4z+2=0=>n_2=(4,2,-4)\\ (n_1,n_2)=12-4-8=0=>\cos(n_1,n_2)=0=>$ planes are perpendicular

c)

$3x+y+2z-1=0=>n_1=(3,1,2)\\ -x+2y+2z+3=0=>n_2=(-1,2,2)\\ (n_1,n_2)=-3+2+4=3\\ |n_1|=\sqrt{14}\quad |n_2|=\sqrt{3}\\ \cos(n_1,n_2)=\frac{\sqrt{3}}{\sqrt{14}}=>$

planes are neither parallel nor perpendicular

d)

$x-3y+2z+1=0=>n_1=(1,-3,2)\\ 3x-4y+2z-1=0=>n_2=(3,-4,2)\\ (n_1,n_2)=3+12+4=19\\ |n_1|=\sqrt{14}\quad |n_2|=\sqrt{29}\\ \cos(n_1,n_2)=\frac{19}{\sqrt{14}\sqrt{29}}\approx0.95=>$

planes are neither parallel nor perpendicular