Question

Determine whether the given planes are parallel or perpendicular and in case they are neither, find the angle between them.

2x- 2y+ 4z+ 5= 0 and 3x- 3y+ 6z- 1= 0

Answer 1

Given planes are

2x-2y+4z+5=0 and 3x-3y+6z-1=0

Here, $a_1=2,b_1=-2,c_1=4and{a}_2=3,b_2=-3,c_2=6$

$\therefore a_1{a}_2+b_1{b}_2+c_1{c}_2=2\times 3+(-2)\times (-3)+4\times 6$

$6+6+24=36\ne 0$

Therefore, the given planes are not perpendicular.

Here $\frac{a_1}{a_2}=\frac{2}{3},\frac{b_1}{b_2}=\frac{-2}{-3},\frac{c_1}{c_2}=\frac{4}{6}=\frac{2}{3}$

It can be seen that $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Thus , the given planes are parallel to each other.