What are the equations of the planes that are parallel to the plane x+2y−2z=1 and two units away from it?
1 Answer
Explanation:
First we'll find the equation of ALL planes parallel to the original one.
As a model consider this lesson:
Equation of a plane parallel to other
The normal vector is:
The equation of the plane parallel to the original one passing through
→n⋅< x−x0,y−y0,z−z0>=0
<1,2,−2>⋅<x−x0,y−y0,z−z0>=0
x−x0+2y−2y0−2z+2z0=0
x+2y−2z−x0−2y0+2z0=0
Or
x+2y−2z+d=0 [1]
wherea=1 ,b=2 ,c=−2 andd=−x0−2y0+2z0
Now we'll find planes that obey the previous formula and at a distance of 2 units from a point in the original plane. (We should expect 2 results, one for each half-space delimited by the original plane.)
As a model consider this lesson:
Distance between 2 parallel planes
In the original plane let's choose a point.
For instance, when
In the formula of the distance between a point and a plane (not any plane but a plane parallel to the original one, equation [1] ), keeping
D=|ax1+by1+cz1+d|√a2+b2+c2
2=∣∣1⋅0+2⋅0+(−2)⋅(−12)+d∣∣√1+4+4
|d+1|=2⋅3 =>|d+1|=6 First solution:
d+1=6 =>d=5
→x+2y−2z+5=0 Second solution:
d+1=−6 =>d=−7
→x+2y−2z−7=0
