# Distance Between Two Planes: Formula and Examples

TL;DR

The distance between two parallel planes \( ax + by + cz + d_1 = 0 \) and \( ax + by + cz + d_2 = 0 \) is \( \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \), provided their normal coefficients match.

## What Is The Distance Between Two Planes?

The **distance between two planes** is the shortest, perpendicular gap between them - but it is only a meaningful, non-zero number when the two planes are **parallel**. If the planes are not parallel, they cross somewhere in space, so the shortest distance between them is **zero**. The distance is measured along the direction of the **normal vector**, the direction pointing straight out of each plane.

So the first question is always: are the two planes parallel? Only then does a distance formula apply.

## When Are Two Planes Parallel?

Write the two planes as:

P1: \( a_1 x + b_1 y + c_1 z + d_1 = 0 \)  
P2: \( a_2 x + b_2 y + c_2 z + d_2 = 0 \)

The vector \( (a,b,c) \) is the plane's **normal** - the direction perpendicular to the surface. Two planes are **parallel** exactly when their normals point the same way, which means the coefficients are proportional:

\( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

If these ratios are all equal, the planes are parallel and a distance can be found. If any ratio differs, the planes tilt relative to each other, they intersect, and the distance is zero. Always check this ratio first.

## The Distance Formula for Parallel Planes

Once you know the planes are parallel, rewrite them so their normal coefficients are **identical**:

P1: \( ax + by + cz + d_1 = 0 \)  
P2: \( ax + by + cz + d_2 = 0 \)

Then the perpendicular distance between them is:

\( d = \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \)

Reading the formula:
- \(|d_1 - d_2|\) is the absolute difference of the two constant terms - the shift between the planes.
- \(\sqrt{a^2 + b^2 + c^2}\) is the length (magnitude) of the shared normal vector \( (a,b,c) \).

The formula is really just the **distance between point and plane** applied to any point on one plane, measured to the other.

## Examples Of Distance Between Two Planes

### Example 1
**Find the distance between the parallel planes \( 2x - y + 2z + 3 = 0 \) and \( 2x - y + 2z + 9 = 0 \).**

The coefficients already match, so \( d_1 = 3 \), \( d_2 = 9 \), and \( (a,b,c)=(2,-1,2) \):

\( d = \frac{|3 - 9|}{\sqrt{2^2 + (-1)^2 + 2^2}} = \frac{6}{\sqrt{4 + 1 + 4}} = \frac{6}{3} = 2 \text{ units} \)

### Example 2
**Find the distance between \( x + 2y + 2z - 6 = 0 \) and \( x + 2y + 2z + 3 = 0 \).**

Coefficients match, \( d_1 = -6 \), \( d_2 = 3 \), normal \( (1,2,2) \):

\( d = \frac{|-6 - 3|}{\sqrt{1^2 + 2^2 + 2^2}} = \frac{9}{\sqrt{1 + 4 + 4}} = 3 \text{ units} \)

### Example 3: The tempting shortcut that misfires
**A student is asked for the distance between \( x + y + z = 1 \) and \( 2x + 3y + z = 5 \)**, and plugs straight into the formula:

\( d = \frac{|1 - 5|}{\sqrt{1 + 1 + 1}} = \frac{4}{\sqrt{3}} \)

The number looks clean, so it feels like the answer. But the formula was applied without the required first check. The correct answer is \( d = 0 \).

### Example 4
**Find the distance between \( 3x - 6y + 2z + 4 = 0 \) and \( 3x - 6y + 2z - 10 = 0 \).**

Matched coefficients, \( d_1 = 4 \), \( d_2 = -10 \):

\( d = \frac{|4 - (-10)|}{\sqrt{3^2 + (-6)^2 + 2^2}} = 2 \text{ units} \)

### Example 5
**Find the distance between \( x - 2y + 2z = 1 \) and \( 2x - 4y + 4z = 10 \).**

The coefficients are not identical, but they are proportional, so the planes **are** parallel:

\( d = \frac{|-1 - (-5)|}{\sqrt{1^2 + (-2)^2 + 2^2}} = \frac{4}{3} \text{ units} \)

### Example 6
**Find the distance between planes \( z = 2 \) and \( z = 7 \).**

\( d = 5 \text{ units} \)

## Where The Distance Between Planes Earns Its Keep

Parallel-plane distance is a working tool in any field that deals with flat layers in three dimensions.

1. **Engineering and manufacturing.**  
2. **Crystallography.**  
3. **3D graphics and games.**

## The Mistakes Students Make Most Often
### Mistake 1: Skipping the parallel check
### Mistake 2: Forgetting to match the coefficients first
### Mistake 3: Mishandling the constant's sign

## Conclusion
- The distance between two parallel planes is \( \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}} \).
- Two planes have a non-zero distance only when they are **parallel**; intersecting planes are zero apart.
- Check parallelism first via the ratios, then scale to match coefficients.