Distance Between Planes Calculator

Distance Between Planes Formula:

\[ \text{Distance} = \frac{|a x_0 + b y_0 + c z_0 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Coefficient a:

Coefficient b:

Coefficient c:

Coefficient d:

Point x-coordinate:

Point y-coordinate:

Point z-coordinate:

Distance Between Planes:

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1. What is Distance Between Planes?

The distance between planes refers to the shortest perpendicular distance from a point to a plane in three-dimensional space. This calculation is fundamental in geometry, physics, and engineering applications involving spatial relationships.

2. How Does the Calculator Work?

The calculator uses the distance formula:

\[ \text{Distance} = \frac{|a x_0 + b y_0 + c z_0 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Where:

\( a, b, c \) — Coefficients of the plane equation \( ax + by + cz + d = 0 \)
\( d \) — Constant term in the plane equation
\( x_0, y_0, z_0 \) — Coordinates of the point

Explanation: The formula calculates the perpendicular distance by projecting the vector from the point to the plane onto the normal vector of the plane.

3. Importance of Distance Calculation

Details: Calculating distances between points and planes is crucial in computer graphics, robotics navigation, architectural design, and physics simulations where spatial relationships determine system behavior.

4. Using the Calculator

Tips: Enter the coefficients a, b, c, d from the plane equation \( ax + by + cz + d = 0 \), and the coordinates of the point. Ensure at least one of a, b, or c is non-zero for a valid plane.

5. Frequently Asked Questions (FAQ)

Q1: What if all coefficients a, b, c are zero?
A: The equation would not represent a valid plane. At least one of a, b, or c must be non-zero.

Q2: Can this calculate distance between two parallel planes?
A: Yes, by using a point from one plane and the equation of the other plane.

Q3: What units are used for the distance?
A: The distance is in the same units as the input coordinates (meters, feet, etc.).

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for the given inputs, limited only by numerical precision.

Q5: Can this be used for 2D distance calculations?
A: For 2D distance from a point to a line, use the 2D version of the formula without the z-component.