0
$\begingroup$

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


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 $$||\overrightarrow{KM}||=||(\mu -\lambda ) \overrightarrow{v}||=|\lambda -\mu | ||\overrightarrow{v}||=\frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}$$

$$$$

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 $\overrightarrow{OM}$ ??
$\endgroup$

1 Answer 1

1
$\begingroup$
  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.

$\endgroup$
1
  • $\begingroup$ I understand!! Thank you so much!!! :-) $\endgroup$
    – Mary Star
    Mar 1, 2015 at 3:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.