Equal spacing between parallel planes

The plane p1p_1 has equation x+y−2z=4x+y-2z=4. A plane p2p_2 with equation 2x+ay+bz=−42x+ay+bz=-4 (where a,ba,b are constants) is parallel to p1p_1, find the values of a,ba,b and also the exact distance between p1,p2p_1,p_2. marks

The plane p3p_3 is parallel to p1,p2p_1,p_2. Find the equation of p3p_3 such that p2p_2 is exactly in between p1,p3p_1,p_3. marks

[modified 2010/NYJC/Prelim/II/4]

Total marks: 5

Solution

Parallel planes

Parallel planes

p1:x+y−2z=4p_1:x+y-2z=4 and p2:2x+ay+bz=−4p_2:2x+ay+bz=-4 are parallel. Find aa and bb.

Explore this idea

Hint

  1. Parallel planes have proportional normals.

Answer · Parallel planes

a=2,b=−4a=2,b=-4.

Explore the idea

p₁ and p₂ are parallel

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

Distance

Distance between parallel planes

Find the exact distance between p1p_1 and p2p_2.

Explore this idea

Hint

  1. Write both planes with the same unit normal 16(1,1,−2)\frac1{\sqrt6}(1,1,-2).

Answer · Distance

6\sqrt6.

Explore the idea

The gap between the planes

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.

Third plane

Equal spacing

Find p3p_3 parallel to both such that p2p_2 is exactly midway between p1p_1 and p3p_3.

Explore this idea

Hint

  1. Along the normal p1p_1 is at 46\frac4{\sqrt6} and p2p_2 at −26-\frac2{\sqrt6}.

  2. p3p_3 is the same distance beyond p2p_2.

Answer · Third plane

x+y−2z=−8x+y-2z=-8.

Explore the idea

Equally spaced planes

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

Open the interactive view to explore this geometry in 3D.