Two parallel planes

Question

When two planes have the same perpendicular vector $\overrightarrow{v}$, then they are parallel, right??

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We have that the two planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$ are parallel, since they have the same perpendicular vector $\overrightarrow{v}=(A, B, C)$.

The line that contains $\overrightarrow{v}$ intersects the two planes at the points $K, M$.

Then $\overrightarrow{OK}=\lambda \overrightarrow{v}=(\lambda A, \lambda B, \lambda C), \overrightarrow{OM}=\mu \overrightarrow{v}=(\mu A, \mu B, \mu C)$.

Then $\lambda A^2 +\lambda B^2 +\lambda C^2=-D_1 \Rightarrow \lambda=-\frac{D_1}{A^2+B^2+C^2}$

$\mu A^2 +\mu B^2 +\mu C^2=-D_1 \Rightarrow \mu=-\frac{D_2}{A^2+B^2+C^2}$

So, the distance between the two planes is equal to $$||\overrightarrow{KM}||=||(\mu -\lambda ) \overrightarrow{v}||=|\lambda -\mu | ||\overrightarrow{v}||=\frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}$$

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My questions are the following:

1. At the sentence " The line that contains $\overrightarrow{v}$ intersects the two planes at the points $K, M$. " what does it mean the line that contains the vector??
2. Why do we calculate the vectors $\overrightarrow{OK}$ and $\overrightarrow{OM}$ ??

Answer 1

1. By itself "the line that contains the vector $\bf v$" doesn't make a lot of sense. But from the following text it appears that what the writer means is "the line through the origin in the direction of $\bf v$".

2. The line segment $KM$ is a perpendicular to both planes, therefore its length is equal to the distance between the planes. We can find $KM$ as $OM-OK$. That's why we calculate those vectors.