Distance from Point to Plane Calculator

In three-dimensional space, a plane is defined by the linear equation Ax + By + Cz + D = 0, where A, B, and C represent the normal vector coefficients perpendicular to the plane. Given a specific point P with coordinates (x0, y0, z0), the shortest distance (d) to this plane is measured along the line normal to the plane. The mathematical formula used is:

d = |A*x0 + B*y0 + C*z0 + D| / sqrt(A^2 + B^2 + C^2)

The numerator evaluates how far off the point is from satisfying the plane equation, while the denominator normalizes this value using the magnitude of the plane's normal vector. If you are instead given a plane passing through a specific point and defined by a normal vector, an alternative form applies the dot product relative to that known anchor point.

Let us find the shortest distance from the point P(1, 2, 3) to the plane defined by the equation 2x - 2y + z - 5 = 0.

Step 1: Identify the coefficients from the plane equation. Here, A = 2, B = -2, C = 1, and D = -5. Also, identify the point coordinates: x0 = 1, y0 = 2, and z0 = 3.

Step 2: Substitute these values into the numerator of the distance formula: |A(x0) + B(y0) + C(z0) + D| = |(2)(1) + (-2)(2) + (1)(3) + (-5)|.

Step 3: Evaluate the numerator: |2 - 4 + 3 - 5| = |-4| = 4.

Step 4: Compute the denominator, which is the magnitude of the normal vector: sqrt(2^2 + (-2)^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt(9) = 3.

Step 5: Divide the numerator by the denominator: d = 4 / 3, which equals approximately 1.333 units.

When setting up your plane equation, always ensure it is expressed in the standard implicit form (Ax + By + Cz + D = 0) before extracting your coefficients. Pay close attention to negative signs on coefficients like B and C, as dropping a negative sign is the most frequent source of calculation errors. Additionally, remember that distance is always a positive scalar value; absolute value brackets in the numerator ensure your final output is never negative, regardless of which side of the plane the point lies on.