1. ## Quick Question: Planes

Two planes A and B have vector equations $\displaystyle r.(2i+j-2k)=3$ and $\displaystyle r.(2i+j-2k)=9$. Explain why A and B are parallel and hence find the distance between them.

Parallel since the perpendicular direction is equal (2i+j-2k) but I have no idea how to calculate the distance
Meh, too late, no replies. Well, I still don't mind an answer to the question I guess...

2. Originally Posted by Lonehwolf
Meh, too late, no replies. Well, I still don't mind an answer to the question I guess...
For both equations $\displaystyle |\vec n| = 3$.
Re-write both equations into Hesse normal form:

$\displaystyle \vec r \cdot (2, 1, -2) -3=0~\implies~\vec r \cdot \dfrac{(2,1, -2)}3 - 1=0$

$\displaystyle \vec r \cdot (2, 1, -2) -9=0~\implies~\vec r \cdot \dfrac{(2,1, -2)}3 - 3=0$

The constant summand equals the distance of the origin to the plane. The difference of those distances equals the distance between the planes:

$\displaystyle d_{planes} = 3-1 = 2$