Build the normal vector and define the line. Supports three-point planes and vector inputs directly. Export results, verify steps, and reuse your values later.
Pick a plane method, then provide the point where the perpendicular line should pass.
Use this sample to test your setup and exports.
| Plane input | Point for line | Perpendicular line (parametric) | Intersection point | Distance point→plane |
|---|---|---|---|---|
| 2x + 3y − 4z + 6 = 0 | (1, −2, 3) | x = 1 + 2t, y = −2 + 3t, z = 3 − 4t | (49/29, −28/29, 47/29) | 10/√29 |
If the line passes through P0 = (x0, y0, z0), then its direction is the plane normal:
A concise reference for perpendicular lines, intersections, and point-to-plane distance.
A line is perpendicular to a plane when its direction matches the plane’s normal vector. In 3D design, this gives a “straight out” direction from a surface, not a tilted projection. The calculator builds that direction from your plane input and anchors it at your chosen point. Typical uses include drilling guides, camera rays, and extrusion directions quickly.
You can define the plane by coefficients (A, B, C, D), by three non‑collinear points, or by one plane point plus two in‑plane vectors. These options cover survey grids, CAD vertices, and engineering drawings where the plane is implied by geometry rather than an explicit equation. When you start with points or vectors, the tool computes A, B, C, then D automatically.
Once the normal n=(A,B,C) is known, the perpendicular line through P0=(x0,y0,z0) is r(t)=P0+t·n. The tool prints parametric and symmetric forms so you can plug into solvers, plot a segment, or verify with substitutions. If a component is zero, the symmetric form is adjusted safely to avoid division by zero.
The intersection point computed is the orthogonal foot from P0 to the plane along the normal direction. The parameter is t=−(A x0+B y0+C z0+D)/(A²+B²+C²). Substituting t back gives the closest point, useful for offsets, standoff distances, and shortest‑path checks. The intersection also satisfies Ax+By+Cz+D=0 by rounding.
The shortest distance from the point to the plane is d=|A x0+B y0+C z0+D|/√(A²+B²+C²). The unit normal u=n/||n|| is also returned so you can express direction cosines, compute angles, or normalize for consistent scaling in simulations and toolpaths.
If A, B, and C are all zero, the plane is invalid and results are blocked. For three‑point input, collinearity is detected using the cross‑product magnitude. For vector input, parallel vectors are rejected. These checks prevent near‑singular normals that can cause huge rounding errors and unstable intersections.
Use the example table to test a full run, then download CSV for spreadsheets or PDF for sharing. In field work, keep units consistent (meters with meters, not mixed). In modeling, store the intersection point and distance as constraints. Repeat with multiple points to map clearance across a surface.
Yes. Enter any point P0, and the tool creates the perpendicular line through P0. It also finds the closest intersection point on the plane and the shortest distance between them.
Choose the three‑point option and enter P1, P2, and P3. The calculator uses a cross product to build the normal. If the points are collinear, it will warn you and stop.
For a plane Ax+By+Cz+D=0, the gradient vector (A, B, C) is perpendicular to every in‑plane direction. Using it as the line direction guarantees a right angle between the line and plane.
The tool substitutes the parametric line into the plane equation and solves for t. The formula is t=−(A x0+B y0+C z0+D)/(A²+B²+C²), which is stable when the normal is nonzero.
Any consistent unit system works. If your coordinates are in meters, the distance output is in meters. Avoid mixing units across x, y, and z or the distance and intersection will be meaningless.
Yes. After a successful calculation, download CSV for spreadsheets or PDF for sharing. Each file includes the plane equation, line forms, intersection point, unit normal, and the point‑to‑plane distance.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.