Question

Find the distance between the planes $2x-y+2z=5$ and $5x-2.5y+5z=20$.

Answer 1

The distance between two parallel planes of the form $a^{}x+b^{}y+c^{}z=d_1,a^{}x+b^{}y+c^{}z=d_2$ is

$\frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}$
$5x-2.5y+5z=20$
On taking $2.5$ common on both sides
$2.5(2x-y+2z)=2.5\times 8$
The second equation can be written as $2x-y+2z=8$

$\Rightarrow$ the distance between the given planes $2x-y+2z=5$ and $2x-y+2z=8$ is $\frac{|8-5|}{\sqrt{2^2+1^2+2^2}}=1$