How to use
- Enter P using its x, y and z coordinates.
- Enter a, b, c and d for ax + by + cz + d = 0. Move every term to the left first.
- Calculate, then compare P with H, the foot of the perpendicular, in the drawing.
Choose one length unit for all coordinates. Changing the unit label does not convert the numbers. Enter numbers, including scientific notation such as 1e-4; do not enter algebraic expressions.
The drawing updates as soon as the inputs describe valid geometry. If a value is missing or invalid, the drawing stays hidden until the inputs are ready. Editing clears the previous numerical result and disables Copy result. Press Calculate to confirm the current numbers, then copy them as text. Clear empties the numerical inputs while keeping your selections.
Formula and interpretation
The shortest path meets the plane at a right angle. Its direction is parallel to n = (a, b, c). Divide the absolute plane expression at P by the normal length to obtain a nonnegative distance. H is the closest point on the plane.
D = |aPₓ + bPᵧ + cP𝓏 + d| / √(a² + b² + c²). H = P − [(aPₓ + bPᵧ + cP𝓏 + d)/(a² + b² + c²)](a,b,c).
Worked example
For P = (2, 3, 7) and z − 2 = 0, enter a = 0, b = 0, c = 1 and d = −2. H = (2, 3, 2), and the distance is 5 units. The dashed segment connects those two points.
A tilted plane gives the same kind of perpendicular distance. For P = (1, 2, 3) and x + 2y + 2z − 3 = 0, the numerator is 8 and the normal length is 3. The distance is 8/3 ≈ 2.67 units, with H = (1/9, 2/9, 11/9). The coordinates shown in the result are rounded; substitution should use the unrounded values.
Uses and limits
Use this for coordinate geometry exercises or the clearance between a point and an ideal flat surface. It does not measure distance to a bounded triangular face: the foot may lie outside that face.
Numbers are calculated with standard floating-point arithmetic. Ordinary results show up to two decimal places; small nonzero values use scientific notation. Equation coefficients use up to 12 significant digits, so a displayed equation can be approximate. Inputs are limited to zero or magnitudes from 10⁻⁹ to 10⁹, and calculated magnitudes to 10¹⁵.
The drawing is an orthographic view, automatically fitted around the geometry. Perspective on the screen does not preserve every angle or length. The small axis arrows indicate orientation, not the location of the origin; read coordinates and distances from the results.
Frequently asked questions
Why is the distance zero?
P already lies on the plane. P and H coincide, so the perpendicular segment has zero length.
Does multiplying the plane equation change the answer?
Multiplying all four coefficients by the same nonzero factor describes the same plane and leaves the distance unchanged. Changing only one coefficient usually changes the plane.
Why is a = b = c = 0 rejected?
A constant expression alone cannot define a unique plane. Supply a nonzero normal.
How should I use the copied numbers?
Paste the result into notes or a worksheet as a calculation check. Keep the coordinate unit and original inputs with it. Rounded displayed coordinates are for reading, not exact symbolic substitution. If a calculation is rejected for precision, shift or rescale the coordinates; do not interpret the empty result as zero.