Point to Plane Distance Calculator

Compute the distance from a point (x0,y0,z0) to the plane ax+by+cz+d=0, with the foot of the perpendicular and the unit normal.

Input

Compute the distance between a point in space and the plane ax + by + cz + d = 0. Enter the point coordinates and the plane coefficients.

Point coordinates

Plane coefficients

Plane equation ax + by + cz + d = 0

Distance from point to plane

Signed distance

Foot of perpendicular (closest point)

(0.333333, 2.333333, 2.333333)

Unit normal vector

(0.666667, -0.333333, 0.666667)

Normal vector length

A positive signed distance means the point lies on the side toward the normal vector, a negative value means the opposite side.

The distance from a point (x0, y0, z0) to the plane ax + by + cz + d = 0 equals the absolute value of a x0 + b y0 + c z0 + d divided by the square root of a² + b² + c².
Removing the absolute value gives the signed distance, taking the normal vector (a, b, c) as the positive direction. It tells you on which side of the plane the point lies.
The closest point on the plane, called the foot of the perpendicular, is found by moving from the point along the unit normal by the signed distance.
If the normal vector (a, b, c) is entirely zero the plane is undefined, so no distance can be computed.

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