Enter Plane Coefficients
Use the standard form ax + by + cz + d = 0.
Plane 2: 4x + 8y + 12z + 20 = 0
Plotly Graph
This graph compares normalized constant terms and the final distance.
Formula Used
For parallel planes, the normal vectors must be proportional.
Plane 1: a₁x + b₁y + c₁z + d₁ = 0
Plane 2: a₂x + b₂y + c₂z + d₂ = 0
If plane 2 coefficients equal k(a₁, b₁, c₁), normalize plane 2 first.
Normalized second plane:
a₁x + b₁y + c₁z + (d₂ / k) = 0
Distance formula:
Distance = |(d₂ / k) - d₁| / √(a₁² + b₁² + c₁²)
Special case: If normals already match, distance = |d₂ - d₁| / √(a² + b² + c²)
How to Use This Calculator
- Enter coefficients for the first plane.
- Enter coefficients for the second plane.
- Keep both equations in standard plane form.
- Click the calculate button.
- Read the distance and normalized plane output.
- Use the graph for quick comparison.
- Download CSV or PDF for records.
Example Data Table
| Example | Plane 1 | Plane 2 | Distance |
|---|---|---|---|
| 1 | x + 2y + 2z + 4 = 0 | x + 2y + 2z + 10 = 0 | 2.000000 |
| 2 | 2x + 4y + 6z + 8 = 0 | 4x + 8y + 12z + 20 = 0 | 0.801784 |
| 3 | 3x + y + 2z - 5 = 0 | 6x + 2y + 4z + 7 = 0 | 2.405351 |
| 4 | 5x - y + 2z + 9 = 0 | 10x - 2y + 4z - 3 = 0 | 1.061913 |
FAQs
1. What does this calculator compute?
It computes the shortest perpendicular distance between two parallel planes in three dimensions. It also checks whether the normal vectors are proportional before showing the final result.
2. Why must the planes be parallel?
A constant separation exists only for parallel planes. Intersecting planes meet along a line, so one unique perpendicular distance between complete planes does not apply.
3. What form should the equations use?
Use the standard form ax + by + cz + d = 0. The calculator reads coefficients directly from this equation and then performs the parallel check and distance calculation.
4. What happens if normals are scaled differently?
That is acceptable. Parallel planes often have scaled normals. The calculator finds the scale factor, normalizes the second plane, and then applies the correct distance formula.
5. Can this tool detect invalid inputs?
Yes. It warns you when a plane has a zero normal vector or when both planes fail the parallel condition. Those cases prevent a valid distance result.
6. Why is normalization important?
Normalization converts both equations to the same normal direction. That makes the constant terms directly comparable and avoids errors when one plane is written as a scaled equation.
7. What units does the answer use?
The answer uses the same length units implied by your coordinate system. If your coordinates are in meters, the distance result is also in meters.
8. Can I export the result?
Yes. Use the CSV button for tabular output or the PDF button for a clean report. Both downloads use your latest calculation values.