1. What is the Distance Between Planes?

The distance between two parallel planes is the shortest perpendicular distance between them. This calculator computes the distance between two planes given in standard form: ax + by + cz + d = 0.

2. How Does the Calculator Work?

The calculator uses the distance between planes formula:

Where:

• \( a, b, c \) — Coefficients of x, y, z in both plane equations
• \( d_1 \) — Constant term of first plane equation
• \( d_2 \) — Constant term of second plane equation

Note: The formula only works for parallel planes. If the planes are not parallel, the distance is zero where they intersect or undefined if they're identical.

3. Importance of Plane Distance Calculation

Applications: Calculating distance between planes is important in 3D geometry, computer graphics, architectural design, and physics simulations where spatial relationships between surfaces need to be determined.

4. Using the Calculator

Instructions: Enter the coefficients (a, b, c) which must be identical for both planes (or proportional for parallel planes), and the constant terms (d₁, d₂) for each plane. The calculator will compute the distance.

5. Frequently Asked Questions (FAQ)

Q1: What if the planes are not parallel?
A: The calculator will return "Undefined" as non-parallel planes either intersect (distance = 0 along the line of intersection) or are skew.

Q2: How do I know if planes are parallel?
A: Planes are parallel if their normal vectors (a,b,c) are scalar multiples of each other.

Q3: What's the distance between identical planes?
A: The distance is zero since they are the same plane.

Q4: Can I use this for planes in different forms?
A: You must first convert plane equations to standard form (ax + by + cz + d = 0).

Q5: What units does the distance use?
A: The distance is in whatever units your plane coefficients are using (consistent units must be used for all inputs).