The Distance from a Point to a Plane
Simple Explanation
The shortest distance from a point (x₀,y₀,z₀) to a plane ax+by+cz=d is found directly from the plane's equation: distance = |ax₀+by₀+cz₀−d| / √(a²+b²+c²).
Why Do We Need It?
This gives the shortest (perpendicular) distance directly from a formula, without needing to find the actual closest point on the plane first.
Formula
Distance from a Point to a Plane
distance = |ax₀+by₀+cz₀−d| / √(a²+b²+c²)
The shortest (perpendicular) distance from a point (x₀,y₀,z₀) to a plane ax+by+cz=d.
- (x₀,y₀,z₀)
- — the point whose distance to the plane is being found
- a, b, c, d
- — the coefficients of the plane's equation, ax+by+cz=d
When to use it: Whenever the shortest distance from a point to a plane is needed.
Worked Example
Find the distance from a point to a plane
Find the distance from the origin (0,0,0) to the plane x+2y+2z=3.
Why Does This Work?
The numerator, |ax₀+by₀+cz₀−d|, measures how far the point is from satisfying the plane's equation exactly — and dividing by √(a²+b²+c²) (the magnitude of the normal vector) rescales that raw measurement into an actual physical (perpendicular) distance.
Real-Life Example
Drone flight-path clearance planning
A drone's flight-planning software needs to calculate how close the drone will pass to a flat rooftop, modeled as a plane.
This formula gives that clearance distance directly from the drone's coordinates and the rooftop's plane equation.
Practice
Find the distance from the point (0,0,5) to the plane 3x+4y=10 (that is, 3x+4y+0z=10).
HardCommon mistake
Forgetting to take the absolute value of the numerator — distance must always be non-negative, but the raw expression ax₀+by₀+cz₀−d can come out negative.
Quick Review
- distance = |ax₀+by₀+cz₀−d| / √(a²+b²+c²).
- The numerator measures how far the point is from the plane's equation; the denominator rescales it into a true distance.
- Always take the absolute value of the numerator.